There is the solution I used, which is based on Linus Borg answer and works fine with vue.js 2.0.

Vue.directive('click-outside', {
  bind: function (el, binding, vnode) {
    el.clickOutsideEvent = function (event) {
      // here I check that click was outside the el and his children
      if (!(el == event.target || el.contains(event.target))) {
        // and if it did, call method provided in attribute value
        vnode.context[binding.expression](event);
      }
    };
    document.body.addEventListener('click', el.clickOutsideEvent)
  },
  unbind: function (el) {
    document.body.removeEventListener('click', el.clickOutsideEvent)
  },
});

You bind to it using v-click-outside:

<div v-click-outside="doStuff">

Here's a small demo

You can find some more info about custom directives and what el, binding, vnode means in

Add tabindex attribute to your component so that it can be focused and do the following:

<template>
    <div
        @focus="handleFocus"
        @focusout="handleFocusOut"
        tabindex="0"
    >
      SOME CONTENT HERE
    </div>
</template>

<script>
export default {    
    methods: {
        handleFocus() {
            // do something here
        },
        handleFocusOut() {
            // do something here
        }
    }
}
</script>

There are two packages available in the community for this task (both are maintained):

For Vue 3:

This answer is based on MadisonTrash's great answer above but updated to use new Vue 3 syntax.

Vue 3 now uses beforeMount instead of bind, and unmounted instead of unbind (src).

const clickOutside = {
  beforeMount: (el, binding) => {
    el.clickOutsideEvent = event => {
      // here I check that click was outside the el and his children
      if (!(el == event.target || el.contains(event.target))) {
        // and if it did, call method provided in attribute value
        binding.value();
      }
    };
    document.addEventListener("click", el.clickOutsideEvent);
  },
  unmounted: el => {
    document.removeEventListener("click", el.clickOutsideEvent);
  },
};

createApp(App)
  .directive("click-outside", clickOutside)
  .mount("#app");

I did it a slightly different way using a function within created().

  created() {
      window.addEventListener('click', (e) => {
        if (!this.$el.contains(e.target)){
          this.showMobileNav = false
        }
      })
  },

This way, if someone clicks outside of the element, then in my case, the mobile nav is hidden.

I have combined all answers (including a line from vue-clickaway) and came up with this solution that works for me:

Vue.directive('click-outside', {
    bind(el, binding, vnode) {
        var vm = vnode.context;
        var callback = binding.value;

        el.clickOutsideEvent = function (event) {
            if (!(el == event.target || el.contains(event.target))) {
                return callback.call(vm, event);
            }
        };
        document.body.addEventListener('click', el.clickOutsideEvent);
    },
    unbind(el) {
        document.body.removeEventListener('click', el.clickOutsideEvent);
    }
});

Use in component:

<li v-click-outside="closeSearch">
  <!-- your component here -->
</li>

This Worked for me with Vue.js 2.5.2 :

/**
 * Call a function when a click is detected outside of the
 * current DOM node ( AND its children )
 *
 * Example :
 *
 * <template>
 *   <div v-click-outside="onClickOutside">Hello</div>
 * </template>
 *
 * <script>
 * import clickOutside from '../../../../directives/clickOutside'
 * export default {
 *   directives: {
 *     clickOutside
 *   },
 *   data () {
 *     return {
         showDatePicker: false
 *     }
 *   },
 *   methods: {
 *     onClickOutside (event) {
 *       this.showDatePicker = false
 *     }
 *   }
 * }
 * </script>
 */
export default {
  bind: function (el, binding, vNode) {
    el.__vueClickOutside__ = event => {
      if (!el.contains(event.target)) {
        // call method provided in v-click-outside value
        vNode.context[binding.expression](event)
        event.stopPropagation()
      }
    }
    document.body.addEventListener('click', el.__vueClickOutside__)
  },
  unbind: function (el, binding, vNode) {
    // Remove Event Listeners
    document.body.removeEventListener('click', el.__vueClickOutside__)
    el.__vueClickOutside__ = null
  }
}

If you're specifically looking for a click outside the element but still within the parent, you can use

<div class="parent" @click.self="onParentClick">
  <div class="child"></div>
</div>

I use this for modals.

Vue 3 has breaking changes in directives, all of <Vue3 methods were changed/updated. If you wonder, how to do it in Vue 3, Here's the snippet. For information please go through this link

<div v-click-outside="methodToInvoke"></div>

click-outside.js

export default {
  beforeMount: function (el, binding, vnode) {
    binding.event = function (event) {
      if (!(el === event.target || el.contains(event.target))) {
        if (binding.value instanceof Function) {
          binding.value(event)
        }
      }
    }
    document.body.addEventListener('click', binding.event)
  },
  unmounted: function (el, binding, vnode) {
    document.body.removeEventListener('click', binding.event)
  }
}

and In main.js add the following

// Directives
import ClickOutside from './click-outside'

createApp(App)
 .directive('click-outside', ClickOutside)
 .use(IfAnyModules)
 .mount('#app')

I have updated MadisonTrash's answer to support Mobile Safari (which does not have click event, touchend must be used instead). This also incorporates a check so that the event isn't triggered by dragging on mobile devices.

Vue.directive('click-outside', {
    bind: function (el, binding, vnode) {
        el.eventSetDrag = function () {
            el.setAttribute('data-dragging', 'yes');
        }
        el.eventClearDrag = function () {
            el.removeAttribute('data-dragging');
        }
        el.eventOnClick = function (event) {
            var dragging = el.getAttribute('data-dragging');
            // Check that the click was outside the el and its children, and wasn't a drag
            if (!(el == event.target || el.contains(event.target)) && !dragging) {
                // call method provided in attribute value
                vnode.context[binding.expression](event);
            }
        };
        document.addEventListener('touchstart', el.eventClearDrag);
        document.addEventListener('touchmove', el.eventSetDrag);
        document.addEventListener('click', el.eventOnClick);
        document.addEventListener('touchend', el.eventOnClick);
    }, unbind: function (el) {
        document.removeEventListener('touchstart', el.eventClearDrag);
        document.removeEventListener('touchmove', el.eventSetDrag);
        document.removeEventListener('click', el.eventOnClick);
        document.removeEventListener('touchend', el.eventOnClick);
        el.removeAttribute('data-dragging');
    },
});

export default {
  bind: function (el, binding, vNode) {
    // Provided expression must evaluate to a function.
    if (typeof binding.value !== 'function') {
      const compName = vNode.context.name
      let warn = `[Vue-click-outside:] provided expression '${binding.expression}' is not a function, but has to be`
      if (compName) { warn += `Found in component '${compName}'` }

      console.warn(warn)
    }
    // Define Handler and cache it on the element
    const bubble = binding.modifiers.bubble
    const handler = (e) => {
      if (bubble || (!el.contains(e.target) && el !== e.target)) {
        binding.value(e)
      }
    }
    el.__vueClickOutside__ = handler

    // add Event Listeners
    document.addEventListener('click', handler)
  },

  unbind: function (el, binding) {
    // Remove Event Listeners
    document.removeEventListener('click', el.__vueClickOutside__)
    el.__vueClickOutside__ = null

  }
}

Complete case for vue 3

This is a complete solution based on MadisonTrash answer, and benrwb and fredrivett tweaks for safari compatibility and vue 3 api changes.

Edit:

The solution proposed below is still useful, and the how to use is still valid, but I changed it to use document.elementsFromPoint instead of event.contains because it doesn't recognise as children some elements like the <path> tags inside svgs. So the right directive is this one:

export default {
    beforeMount: (el, binding) => {
        el.eventSetDrag = () => {
            el.setAttribute("data-dragging", "yes");
        };
        el.eventClearDrag = () => {
            el.removeAttribute("data-dragging");
        };
        el.eventOnClick = event => {
            const dragging = el.getAttribute("data-dragging");
            // Check that the click was outside the el and its children, and wasn't a drag
            console.log(document.elementsFromPoint(event.clientX, event.clientY))
            if (!document.elementsFromPoint(event.clientX, event.clientY).includes(el) && !dragging) {
                // call method provided in attribute value
                binding.value(event);
            }
        };
        document.addEventListener("touchstart", el.eventClearDrag);
        document.addEventListener("touchmove", el.eventSetDrag);
        document.addEventListener("click", el.eventOnClick);
        document.addEventListener("touchend", el.eventOnClick);
    },
    unmounted: el => {
        document.removeEventListener("touchstart", el.eventClearDrag);
        document.removeEventListener("touchmove", el.eventSetDrag);
        document.removeEventListener("click", el.eventOnClick);
        document.removeEventListener("touchend", el.eventOnClick);
        el.removeAttribute("data-dragging");
    },
};