What Is a Monad?

First: The term monad is a bit vacuous if you are not a mathematician. An alternative term is computation builder which is a bit more descriptive of what they are actually useful for.

They are a pattern for chaining operations. It looks a bit like method chaining in object-oriented languages, but the mechanism is slightly different.

The pattern is mostly used in functional languages (especially Haskell which uses monads pervasively) but can be used in any language which support higher-order functions (that is, functions which can take other functions as arguments).

Arrays in JavaScript support the pattern, so let’s use that as the first example.

The gist of the pattern is we have a type (Array in this case) which has a method which takes a function as argument. The operation supplied must return an instance of the same type (i.e. return an Array).

First an example of method chaining which does not use the monad pattern:

[1,2,3].map(x => x + 1)

The result is [2,3,4]. The code does not conform to the monad pattern, since the function we are supplying as an argument returns a number, not an Array. The same logic in monad form would be:

[1,2,3].flatMap(x => [x + 1])

Here we supply an operation which returns an Array, so now it conforms to the pattern. The flatMap method executes the provided function for every element in the array. It expects an array as result for each invocation (rather than single values), but merges the resulting set of arrays into a single array. So the end result is the same, the array [2,3,4].

(The function argument provided to a method like map or flatMap is often called a "callback" in JavaScript. I will call it the "operation" since it is more general.)

If we chain multiple operations (in the traditional way):

[1,2,3].map(a => a + 1).filter(b => b != 3)

Results in the array [2,4]

The same chaining in monad form:

[1,2,3].flatMap(a => [a + 1]).flatMap(b => b != 3 ? [b] : [])

Yields the same result, the array [2,4].

You will immediately notice that the monad form is quite a bit uglier than the non-monad! This just goes to show that monads are not necessarily “good”. They are a pattern which is sometimes beneficial and sometimes not.

Do note that the monad pattern can be combined in a different way:

[1,2,3].flatMap(a => [a + 1].flatMap(b => b != 3 ? [b] : []))

Here the binding is nested rather than chained, but the result is the same. This is an important property of monads as we will see later. It means two operations combined can be treated the same as a single operation.

The operation is allowed to return an array with different element types, for example transforming an array of numbers into an array of strings or something else; as long as it still an Array.

This can be described a bit more formally using Typescript notation. An array has the type Array<T>, where T is the type of the elements in the array. The method flatMap() takes a function argument of the type T => Array<U> and returns an Array<U>.

Generalized, a monad is any type Foo<Bar> which has a "bind" method which takes a function argument of type Bar => Foo<Baz> and returns a Foo<Baz>.

This answers what monads are. The rest of this answer will try to explain through examples why monads can be a useful pattern in a language like Haskell which has good support for them.

Haskell and Do-notation

To translate the map/filter example directly to Haskell, we replace flatMap with the >>= operator:

[1,2,3] >>= \a -> [a+1] >>= \b -> if b == 3 then [] else [b] 

The >>= operator is the bind function in Haskell. It does the same as flatMap in JavaScript when the operand is a list, but it is overloaded with different meaning for other types.

But Haskell also has a dedicated syntax for monad expressions, the do-block, which hides the bind operator altogether:

 do a <- [1,2,3] 
    b <- [a+1] 
    if b == 3 then [] else [b] 

This hides the "plumbing" and lets you focus on the actual operations applied at each step.

In a do-block, each line is an operation. The constraint still holds that all operations in the block must return the same type. Since the first expression is a list, the other operations must also return a list.

The back-arrow <- looks deceptively like an assignment, but note that this is the parameter passed in the bind. So, when the expression on the right side is a List of Integers, the variable on the left side will be a single Integer – but will be executed for each integer in the list.

Example: Safe navigation (the Maybe type)

Enough about lists, lets see how the monad pattern can be useful for other types.

Some functions may not always return a valid value. In Haskell this is represented by the Maybe-type, which is an option that is either Just value or Nothing.

Chaining operations which always return a valid value is of course straightforward:

streetName = getStreetName (getAddress (getUser 17)) 

But what if any of the functions could return Nothing? We need to check each result individually and only pass the value to the next function if it is not Nothing:

case getUser 17 of
      Nothing -> Nothing 
      Just user ->
         case getAddress user of
            Nothing -> Nothing 
            Just address ->
              getStreetName address

Quite a lot of repetitive checks! Imagine if the chain was longer. Haskell solves this with the monad pattern for Maybe:

do
  user <- getUser 17
  addr <- getAddress user
  getStreetName addr

This do-block invokes the bind-function for the Maybe type (since the result of the first expression is a Maybe). The bind-function only executes the following operation if the value is Just value, otherwise it just passes the Nothing along.

Here the monad-pattern is used to avoid repetitive code. This is similar to how some other languages use macros to simplify syntax, although macros achieve the same goal in a very different way.

Note that it is the combination of the monad pattern and the monad-friendly syntax in Haskell which result in the cleaner code. In a language like JavaScript without any special syntax support for monads, I doubt the monad pattern would be able to simplify the code in this case.

Mutable state

Haskell does not support mutable state. All variables are constants and all values immutable. But the State type can be used to emulate programming with mutable state:

add2 :: State Integer Integer
add2 = do
        -- add 1 to state
         x <- get
         put (x + 1)
         -- increment in another way
         modify (+1)
         -- return state
         get


evalState add2 7
=> 9

The add2 function builds a monad chain which is then evaluated with 7 as the initial state.

Obviously this is something which only makes sense in Haskell. Other languages support mutable state out of the box. Haskell is generally "opt-in" on language features - you enable mutable state when you need it, and the type system ensures the effect is explicit. IO is another example of this.

IO

The IO type is used for chaining and executing “impure” functions.

Like any other practical language, Haskell has a bunch of built-in functions which interface with the outside world: putStrLine, readLine and so on. These functions are called “impure” because they either cause side effects or have non-deterministic results. Even something simple like getting the time is considered impure because the result is non-deterministic – calling it twice with the same arguments may return different values.

A pure function is deterministic – its result depends purely on the arguments passed and it has no side effects on the environment beside returning a value.

Haskell heavily encourages the use of pure functions – this is a major selling point of the language. Unfortunately for purists, you need some impure functions to do anything useful. The Haskell compromise is to cleanly separate pure and impure, and guarantee that there is no way that pure functions can execute impure functions, directly or indirect.

This is guaranteed by giving all impure functions the IO type. The entry point in Haskell program is the main function which have the IO type, so we can execute impure functions at the top level.

But how does the language prevent pure functions from executing impure functions? This is due to the lazy nature of Haskell. A function is only executed if its output is consumed by some other function. But there is no way to consume an IO value except to assign it to main. So if a function wants to execute an impure function, it has to be connected to main and have the IO type.

Using monad chaining for IO operations also ensures that they are executed in a linear and predictable order, just like statements in an imperative language.

This brings us to the first program most people will write in Haskell:

main :: IO ()
main = do 
        putStrLn ”Hello World”

The do keyword is superfluous when there is only a single operation and therefore nothing to bind, but I keep it anyway for consistency.

The () type means “void”. This special return type is only useful for IO functions called for their side effect.

A longer example:

main = do
    putStrLn "What is your name?"
    name <- getLine
    putStrLn "hello" ++ name

This builds a chain of IO operations, and since they are assigned to the main function, they get executed.

Comparing IO with Maybe shows the versatility of the monad pattern. For Maybe, the pattern is used to avoid repetitive code by moving conditional logic to the binding function. For IO, the pattern is used to ensure that all operations of the IO type are sequenced and that IO operations cannot "leak" to pure functions.

Summing up

In my subjective opinion, the monad pattern is only really worthwhile in a language which has some built-in support for the pattern. Otherwise it just leads to overly convoluted code. But Haskell (and some other languages) have some built-in support which hides the tedious parts, and then the pattern can be used for a variety of useful things. Like:

• Avoiding repetitive code (Maybe)
• Adding language features like mutable state or exceptions for delimited areas of the program.
• Isolating icky stuff from nice stuff (IO)
• Embedded domain-specific languages (Parser)
• Adding GOTO to the language.

Explaining "what is a monad" is a bit like saying "what is a number?" We use numbers all the time. But imagine you met someone who didn't know anything about numbers. How the heck would you explain what numbers are? And how would you even begin to describe why that might be useful?

What is a monad? The short answer: It's a specific way of chaining operations together.

In essence, you're writing execution steps and linking them together with the "bind function". (In Haskell, it's named >>=.) You can write the calls to the bind operator yourself, or you can use syntax sugar which makes the compiler insert those function calls for you. But either way, each step is separated by a call to this bind function.

So the bind function is like a semicolon; it separates the steps in a process. The bind function's job is to take the output from the previous step, and feed it into the next step.

That doesn't sound too hard, right? But there is more than one kind of monad. Why? How?

Well, the bind function can just take the result from one step, and feed it to the next step. But if that's "all" the monad does... that actually isn't very useful. And that's important to understand: Every useful monad does something else in addition to just being a monad. Every useful monad has a "special power", which makes it unique.

(A monad that does nothing special is called the "identity monad". Rather like the identity function, this sounds like an utterly pointless thing, yet turns out not to be... But that's another story™.)

Basically, each monad has its own implementation of the bind function. And you can write a bind function such that it does hoopy things between execution steps. For example:

• If each step returns a success/failure indicator, you can have bind execute the next step only if the previous one succeeded. In this way, a failing step aborts the whole sequence "automatically", without any conditional testing from you. (The Failure Monad.)

• Extending this idea, you can implement "exceptions". (The Error Monad or Exception Monad.) Because you're defining them yourself rather than it being a language feature, you can define how they work. (E.g., maybe you want to ignore the first two exceptions and only abort when a third exception is thrown.)

• You can make each step return multiple results, and have the bind function loop over them, feeding each one into the next step for you. In this way, you don't have to keep writing loops all over the place when dealing with multiple results. The bind function "automatically" does all that for you. (The List Monad.)

• As well as passing a "result" from one step to another, you can have the bind function pass extra data around as well. This data now doesn't show up in your source code, but you can still access it from anywhere, without having to manually pass it to every function. (The Reader Monad.)

• You can make it so that the "extra data" can be replaced. This allows you to simulate destructive updates, without actually doing destructive updates. (The State Monad and its cousin the Writer Monad.)

• Because you're only simulating destructive updates, you can trivially do things that would be impossible with real destructive updates. For example, you can undo the last update, or revert to an older version.

• You can make a monad where calculations can be paused, so you can pause your program, go in and tinker with internal state data, and then resume it.

• You can implement "continuations" as a monad. This allows you to break people's minds!

All of this and more is possible with monads. Of course, all of this is also perfectly possible without monads too. It's just drastically easier using monads.

Actually, contrary to common understanding of Monads, they have nothing to do with state. Monads are simply a way to wrapping things and provide methods to do operations on the wrapped stuff without unwrapping it.

For example, you can create a type to wrap another one, in Haskell:

data Wrapped a = Wrap a

To wrap stuff we define

return :: a -> Wrapped a
return x = Wrap x

To perform operations without unwrapping, say you have a function f :: a -> b, then you can do this to lift that function to act on wrapped values:

fmap :: (a -> b) -> (Wrapped a -> Wrapped b)
fmap f (Wrap x) = Wrap (f x)

That's about all there is to understand. However, it turns out that there is a more general function to do this lifting, which is bind:

bind :: (a -> Wrapped b) -> (Wrapped a -> Wrapped b)
bind f (Wrap x) = f x

bind can do a bit more than fmap, but not vice versa. Actually, fmap can be defined only in terms of bind and return. So, when defining a monad.. you give its type (here it was Wrapped a) and then say how its return and bind operations work.

The cool thing is that this turns out to be such a general pattern that it pops up all over the place, encapsulating state in a pure way is only one of them.

For a good article on how monads can be used to introduce functional dependencies and thus control order of evaluation, like it is used in Haskell's IO monad, check out IO Inside.

As for understanding monads, don't worry too much about it. Read about them what you find interesting and don't worry if you don't understand right away. Then just diving in a language like Haskell is the way to go. Monads are one of these things where understanding trickles into your brain by practice, one day you just suddenly realize you understand them.

But, You could have invented Monads!

sigfpe says:

But all of these introduce monads as something esoteric in need of explanation. But what I want to argue is that they aren't esoteric at all. In fact, faced with various problems in functional programming you would have been led, inexorably, to certain solutions, all of which are examples of monads. In fact, I hope to get you to invent them now if you haven't already. It's then a small step to notice that all of these solutions are in fact the same solution in disguise. And after reading this, you might be in a better position to understand other documents on monads because you'll recognise everything you see as something you've already invented.

Many of the problems that monads try to solve are related to the issue of side effects. So we'll start with them. (Note that monads let you do more than handle side-effects, in particular many types of container object can be viewed as monads. Some of the introductions to monads find it hard to reconcile these two different uses of monads and concentrate on just one or the other.)

In an imperative programming language such as C++, functions behave nothing like the functions of mathematics. For example, suppose we have a C++ function that takes a single floating point argument and returns a floating point result. Superficially it might seem a little like a mathematical function mapping reals to reals, but a C++ function can do more than just return a number that depends on its arguments. It can read and write the values of global variables as well as writing output to the screen and receiving input from the user. In a pure functional language, however, a function can only read what is supplied to it in its arguments and the only way it can have an effect on the world is through the values it returns.

A monad is a datatype that has two operations: >>= (aka bind) and return (aka unit). return takes an arbitrary value and creates an instance of the monad with it. >>= takes an instance of the monad and maps a function over it. (You can see already that a monad is a strange kind of datatype, since in most programming languages you couldn't write a function that takes an arbitrary value and creates a type from it. Monads use a kind of parametric polymorphism.)

In Haskell notation, the monad interface is written

class Monad m where
  return :: a -> m a
  (>>=) :: forall a b . m a -> (a -> m b) -> m b

These operations are supposed to obey certain "laws", but that's not terrifically important: the "laws" just codify the way sensible implementations of the operations ought to behave (basically, that >>= and return ought to agree about how values get transformed into monad instances and that >>= is associative).

Monads are not just about state and I/O: they abstract a common pattern of computation that includes working with state, I/O, exceptions, and non-determinism. Probably the simplest monads to understand are lists and option types:

instance Monad [ ] where
    []     >>= k = []
    (x:xs) >>= k = k x ++ (xs >>= k)
    return x     = [x]

instance Monad Maybe where
    Just x  >>= k = k x
    Nothing >>= k = Nothing
    return x      = Just x

where [] and : are the list constructors, ++ is the concatenation operator, and Just and Nothing are the Maybe constructors. Both of these monads encapsulate common and useful patterns of computation on their respective data types (note that neither has anything to do with side effects or I/O).

You really have to play around writing some non-trivial Haskell code to appreciate what monads are about and why they are useful.

You should first understand what a functor is. Before that, understand higher-order functions.

A higher-order function is simply a function that takes a function as an argument.

A functor is any type construction T for which there exists a higher-order function, call it map, that transforms a function of type a -> b (given any two types a and b) into a function T a -> T b. This map function must also obey the laws of identity and composition such that the following expressions return true for all p and q (Haskell notation):

map id = id
map (p . q) = map p . map q

For example, a type constructor called List is a functor if it comes equipped with a function of type (a -> b) -> List a -> List b which obeys the laws above. The only practical implementation is obvious. The resulting List a -> List b function iterates over the given list, calling the (a -> b) function for each element, and returns the list of the results.

A monad is essentially just a functor T with two extra methods, join, of type T (T a) -> T a, and unit (sometimes called return, fork, or pure) of type a -> T a. For lists in Haskell:

join :: [[a]] -> [a]
pure :: a -> [a]

Why is that useful? Because you could, for example, map over a list with a function that returns a list. Join takes the resulting list of lists and concatenates them. List is a monad because this is possible.

You can write a function that does map, then join. This function is called bind, or flatMap, or (>>=), or (=<<). This is normally how a monad instance is given in Haskell.

A monad has to satisfy certain laws, namely that join must be associative. This means that if you have a value x of type [[[a]]] then join (join x) should equal join (map join x). And pure must be an identity for join such that join (pure x) == x.

[Disclaimer: I am still trying to fully grok monads. The following is just what I have understood so far. If it’s wrong, hopefully someone knowledgeable will call me on the carpet.]

Arnar wrote:

Monads are simply a way to wrapping things and provide methods to do operations on the wrapped stuff without unwrapping it.

That’s precisely it. The idea goes like this:

1. You take some kind of value and wrap it with some additional information. Just like the value is of a certain kind (eg. an integer or a string), so the additional information is of a certain kind.

   E.g., that extra information might be a Maybe or an IO.

2. Then you have some operators that allow you to operate on the wrapped data while carrying along that additional information. These operators use the additional information to decide how to change the behaviour of the operation on the wrapped value.

   E.g., a Maybe Int can be a Just Int or Nothing. Now, if you add a Maybe Int to a Maybe Int, the operator will check to see if they are both Just Ints inside, and if so, will unwrap the Ints, pass them the addition operator, re-wrap the resulting Int into a new Just Int (which is a valid Maybe Int), and thus return a Maybe Int. But if one of them was a Nothing inside, this operator will just immediately return Nothing, which again is a valid Maybe Int. That way, you can pretend that your Maybe Ints are just normal numbers and perform regular math on them. If you were to get a Nothing, your equations will still produce the right result – without you having to litter checks for Nothing everywhere.

But the example is just what happens for Maybe. If the extra information was an IO, then that special operator defined for IOs would be called instead, and it could do something totally different before performing the addition. (OK, adding two IO Ints together is probably nonsensical – I’m not sure yet.) (Also, if you paid attention to the Maybe example, you have noticed that “wrapping a value with extra stuff” is not always correct. But it’s hard to be exact, correct and precise without being inscrutable.)

Basically, “monad” roughly means “pattern”. But instead of a book full of informally explained and specifically named Patterns, you now have a language construct – syntax and all – that allows you to declare new patterns as things in your program. (The imprecision here is all the patterns have to follow a particular form, so a monad is not quite as generic as a pattern. But I think that’s the closest term that most people know and understand.)

And that is why people find monads so confusing: because they are such a generic concept. To ask what makes something a monad is similarly vague as to ask what makes something a pattern.

But think of the implications of having syntactic support in the language for the idea of a pattern: instead of having to read the Gang of Four book and memorise the construction of a particular pattern, you just write code that implements this pattern in an agnostic, generic way once and then you are done! You can then reuse this pattern, like Visitor or Strategy or Façade or whatever, just by decorating the operations in your code with it, without having to re-implement it over and over!

So that is why people who understand monads find them so useful: it’s not some ivory tower concept that intellectual snobs pride themselves on understanding (OK, that too of course, teehee), but actually makes code simpler.

After much striving, I think I finally understand the monad. After rereading my own lengthy critique of the overwhelmingly top voted answer, I will offer this explanation.

There are three questions that need to be answered to understand monads:

1. Why do you need a monad?
2. What is a monad?
3. How is a monad implemented?

As I noted in my original comments, too many monad explanations get caught up in question number 3, without, and before really adequately covering question 2, or question 1.

Why do you need a monad?

Pure functional languages like Haskell are different from imperative languages like C, or Java in that, a pure functional program is not necessarily executed in a specific order, one step at a time. A Haskell program is more akin to a mathematical function, in which you may solve the "equation" in any number of potential orders. This confers a number of benefits, among which is that it eliminates the possibility of certain kinds of bugs, particularly those relating to things like "state".

However, there are certain problems that are not so straightforward to solve with this style of programming. Some things, like console programming, and file i/o, need things to happen in a particular order, or need to maintain state. One way to deal with this problem is to create a kind of object that represents the state of a computation, and a series of functions that take a state object as input, and return a new modified state object.

So let's create a hypothetical "state" value, that represents the state of a console screen. exactly how this value is constructed is not important, but let's say it's an array of byte length ascii characters that represents what is currently visible on the screen, and an array that represents the last line of input entered by the user, in pseudocode. We've defined some functions that take console state, modify it, and return a new console state.

consolestate MyConsole = new consolestate;

So to do console programming, but in a pure functional manner, you would need to nest a lot of function calls inside eachother.

consolestate FinalConsole = print(input(print(myconsole, "Hello, what's your name?")),"hello, %inputbuffer%!");

Programming in this way keeps the "pure" functional style, while forcing changes to the console to happen in a particular order. But, we'll probably want to do more than just a few operations at a time like in the above example. Nesting functions in that way will start to become ungainly. What we want, is code that does essentially the same thing as above, but is written a bit more like this:

consolestate FinalConsole = myconsole:
                            print("Hello, what's your name?"):
                            input():
                            print("hello, %inputbuffer%!");

This would indeed be a more convenient way to write it. How do we do that though?

What is a monad?

Once you have a type (such as consolestate) that you define along with a bunch of functions designed specifically to operate on that type, you can turn the whole package of these things into a "monad" by defining an operator like : (bind) that automatically feeds return values on its left, into function parameters on its right, and a lift operator that turns normal functions, into functions that work with that specific kind of bind operator.

How is a monad implemented?

See other answers, that seem quite free to jump into the details of that.

After giving an answer to this question a few years ago, I believe I can improve and simplify that response with...

A monad is a function composition technique that externalizes treatment for some input scenarios using a composing function, bind, to pre-process input during composition.

In normal composition, the function, compose (>>), is use to apply the composed function to the result of its predecessor in sequence. Importantly, the function being composed is required to handle all scenarios of its input.

(x -> y) >> (y -> z)

This design can be improved by restructuring the input so that relevant states are more easily interrogated. So, instead of simply y the value can become Mb such as, for instance, (is_OK, b) if y included a notion of validity.

For example, when the input is only possibly a number, instead of returning a string which can dutifully contain a number or not, you could restructure the type into a bool indicating the presence of a valid number and a number in tuple such as, bool * float. The composed functions would now no longer need to parse an input string to determine whether a number exists but could merely inspect the bool portion of a tuple.

(Ma -> Mb) >> (Mb -> Mc)

Here, again, composition occurs naturally with compose and so each function must handle all scenarios of its input individually, though doing so is now much easier.

However, what if we could externalize the effort of interrogation for those times where handling a scenario is routine. For example, what if our program does nothing when the input is not OK as in when is_OK is false. If that were done then composed functions would not need to handle that scenario themselves, dramatically simplifying their code and effecting another level of reuse.

To achieve this externalization we could use a function, bind (>>=), to perform the composition instead of compose. As such, instead of simply transferring values from the output of one function to the input of another Bind would inspect the M portion of Ma and decide whether and how to apply the composed function to the a. Of course, the function bind would be defined specifically for our particular M so as to be able to inspect its structure and perform whatever type of application we want. Nonetheless, the a can be anything since bind merely passes the a uninspected to the the composed function when it determines application necessary. Additionally, the composed functions themselves no longer need to deal with the M portion of the input structure either, simplifying them. Hence...

(a -> Mb) >>= (b -> Mc) or more succinctly Mb >>= (b -> Mc)

In short, a monad externalizes and thereby provides standard behaviour around the treatment of certain input scenarios once the input becomes designed to sufficiently expose them. This design is a shell and content model where the shell contains data relevant to the application of the composed function and is interrogated by and remains only available to the bind function.

Therefore, a monad is three things:

1. an M shell for holding monad relevant information,
2. a bind function implemented to make use of this shell information in its application of the composed functions to the content value(s) it finds within the shell, and
3. composable functions of the form, a -> Mb, producing results that include monadic management data.

Generally speaking, the input to a function is far more restrictive than its output which may include such things as error conditions; hence, the Mb result structure is generally very useful. For instance, the division operator does not return a number when the divisor is 0.

Additionally, monads may include wrap functions that wrap values, a, into the monadic type, Ma, and general functions, a -> b, into monadic functions, a -> Mb, by wrapping their results after application. Of course, like bind, such wrap functions are specific to M. An example:

let return a = [a]
let lift f a = return (f a)

The design of the bind function presumes immutable data structures and pure functions others things get complex and guarantees cannot be made. As such, there are monadic laws:

Given...

M_ 
return = (a -> Ma)
f = (a -> Mb)
g = (b -> Mc)

Then...

Left Identity  : (return a) >>= f === f a
Right Identity : Ma >>= return    === Ma
Associative    : Ma >>= (f >>= g) === Ma >>= ((fun x -> f x) >>= g)

Associativity means that bind preserves the order of evaluation regardless of when bind is applied. That is, in the definition of Associativity above, the force early evaluation of the parenthesized binding of f and g will only result in a function that expects Ma in order to complete the bind. Hence the evaluation of Ma must be determined before its value can become applied to f and that result in turn applied to g.

A monad is, effectively, a form of "type operator". It will do three things. First it will "wrap" (or otherwise convert) a value of one type into another type (typically called a "monadic type"). Secondly it will make all the operations (or functions) available on the underlying type available on the monadic type. Finally it will provide support for combining its self with another monad to produce a composite monad.

The "maybe monad" is essentially the equivalent of "nullable types" in Visual Basic / C#. It takes a non nullable type "T" and converts it into a "Nullable<T>", and then defines what all the binary operators mean on a Nullable<T>.

Side effects are represented simillarly. A structure is created that holds descriptions of side effects alongside a function's return value. The "lifted" operations then copy around side effects as values are passed between functions.

They are called "monads" rather than the easier-to-grasp name of "type operators" for several reasons:

1. Monads have restrictions on what they can do (see the definiton for details).
2. Those restrictions, along with the fact that there are three operations involved, conform to the structure of something called a monad in Category Theory, which is an obscure branch of mathematics.
3. They were designed by proponents of "pure" functional languages
4. Proponents of pure functional languages like obscure branches of mathematics
5. Because the math is obscure, and monads are associated with particular styles of programming, people tend to use the word monad as a sort of secret handshake. Because of this no one has bothered to invest in a better name.

(See also the answers at What is a monad?)

A good motivation to Monads is sigfpe (Dan Piponi)'s You Could Have Invented Monads! (And Maybe You Already Have). There are a LOT of other monad tutorials, many of which misguidedly try to explain monads in "simple terms" using various analogies: this is the monad tutorial fallacy; avoid them.

As DR MacIver says in Tell us why your language sucks:

So, things I hate about Haskell:

Let’s start with the obvious. Monad tutorials. No, not monads. Specifically the tutorials. They’re endless, overblown and dear god are they tedious. Further, I’ve never seen any convincing evidence that they actually help. Read the class definition, write some code, get over the scary name.

You say you understand the Maybe monad? Good, you're on your way. Just start using other monads and sooner or later you'll understand what monads are in general.

[If you are mathematically oriented, you might want to ignore the dozens of tutorials and learn the definition, or follow lectures in category theory :) The main part of the definition is that a Monad M involves a "type constructor" that defines for each existing type "T" a new type "M T", and some ways for going back and forth between "regular" types and "M" types.]

Also, surprisingly enough, one of the best introductions to monads is actually one of the early academic papers introducing monads, Philip Wadler's Monads for functional programming. It actually has practical, non-trivial motivating examples, unlike many of the artificial tutorials out there.

Monads are to control flow what abstract data types are to data.

In other words, many developers are comfortable with the idea of Sets, Lists, Dictionaries (or Hashes, or Maps), and Trees. Within those data types there are many special cases (for instance InsertionOrderPreservingIdentityHashMap).

However, when confronted with program "flow" many developers haven't been exposed to many more constructs than if, switch/case, do, while, goto (grr), and (maybe) closures.

So, a monad is simply a control flow construct. A better phrase to replace monad would be 'control type'.

As such, a monad has slots for control logic, or statements, or functions - the equivalent in data structures would be to say that some data structures allow you to add data, and remove it.

For example, the "if" monad:

if( clause ) then block

at its simplest has two slots - a clause, and a block. The if monad is usually built to evaluate the result of the clause, and if not false, evaluate the block. Many developers are not introduced to monads when they learn 'if', and it just isn't necessary to understand monads to write effective logic.

Monads can become more complicated, in the same way that data structures can become more complicated, but there are many broad categories of monad that may have similar semantics, but differing implementations and syntax.

Of course, in the same way that data structures may be iterated over, or traversed, monads may be evaluated.

Compilers may or may not have support for user-defined monads. Haskell certainly does. Ioke has some similar capabilities, although the term monad is not used in the language.

My favorite Monad tutorial:

(out of 170,000 hits on a Google search for "monad tutorial"!)

@Stu: The point of monads is to allow you to add (usually) sequential semantics to otherwise pure code; you can even compose monads (using Monad Transformers) and get more interesting and complicated combined semantics, like parsing with error handling, shared state, and logging, for example. All of this is possible in pure code, monads just allow you to abstract it away and reuse it in modular libraries (always good in programming), as well as providing convenient syntax to make it look imperative.

Haskell already has operator overloading[1]: it uses type classes much the way one might use interfaces in Java or C# but Haskell just happens to also allow non-alphanumeric tokens like + && and > as infix identifiers. It's only operator overloading in your way of looking at it if you mean "overloading the semicolon" [2]. It sounds like black magic and asking for trouble to "overload the semicolon" (picture enterprising Perl hackers getting wind of this idea) but the point is that without monads there is no semicolon, since purely functional code does not require or allow explicit sequencing.

This all sounds much more complicated than it needs to. sigfpe's article is pretty cool but uses Haskell to explain it, which sort of fails to break the chicken and egg problem of understanding Haskell to grok Monads and understanding Monads to grok Haskell.

[1] This is a separate issue from monads but monads use Haskell's operator overloading feature.

[2] This is also an oversimplification since the operator for chaining monadic actions is >>= (pronounced "bind") but there is syntactic sugar ("do") that lets you use braces and semicolons and/or indentation and newlines.

I am still new to monads, but I thought I would share a link I found that felt really good to read (WITH PICTURES!!): (no affiliation)

Basically, the warm and fuzzy concept that I got from the article was the concept that monads are basically adapters that allow disparate functions to work in a composable fashion, i.e. be able to string up multiple functions and mix and match them without worrying about inconsistent return types and such. So the BIND function is in charge of keeping apples with apples and oranges with oranges when we're trying to make these adapters. And the LIFT function is in charge of taking "lower level" functions and "upgrading" them to work with BIND functions and be composable as well.

I hope I got it right, and more importantly, hope that the article has a valid view on monads. If nothing else, this article helped whet my appetite for learning more about monads.

I've been thinking of Monads in a different way, lately. I've been thinking of them as abstracting out execution order in a mathematical way, which makes new kinds of polymorphism possible.

If you're using an imperative language, and you write some expressions in order, the code ALWAYS runs exactly in that order.

And in the simple case, when you use a monad, it feels the same -- you define a list of expressions that happen in order. Except that, depending on which monad you use, your code might run in order (like in IO monad), in parallel over several items at once (like in the List monad), it might halt partway through (like in the Maybe monad), it might pause partway through to be resumed later (like in a Resumption monad), it might rewind and start from the beginning (like in a Transaction monad), or it might rewind partway to try other options (like in a Logic monad).

And because monads are polymorphic, it's possible to run the same code in different monads, depending on your needs.

Plus, in some cases, it's possible to combine monads together (with monad transformers) to get multiple features at the same time.

Monads Are Not Metaphors, but a practically useful abstraction emerging from a common pattern, as Daniel Spiewak explains.

tl;dr

{-# LANGUAGE InstanceSigs #-}

newtype Id t = Id t

instance Monad Id where
   return :: t -> Id t
   return = Id

   (=<<) :: (a -> Id b) -> Id a -> Id b
   f =<< (Id x) = f x