I just want to update the answer to match the new React.createRef() method, but it's basically the same, just have in mind the current property in the created ref:

class Messages extends React.Component {

  const messagesEndRef = React.createRef()

  componentDidMount () {
    this.scrollToBottom()
  }
  componentDidUpdate () {
    this.scrollToBottom()
  }
  scrollToBottom = () => {
    this.messagesEndRef.current.scrollIntoView({ behavior: 'smooth' })
  }
  render () {
    const { messages } = this.props
    return (
      <div>
        {messages.map(message => <Message key={message.id} {...message} />)}
        <div ref={this.messagesEndRef} />
      </div>
    )
  }
}

UPDATE:

Now that hooks are available, I'm updating the answer to add the use of the useRef and useEffect hooks, the real thing doing the magic (React refs and scrollIntoView DOM method) remains the same:

import React, { useEffect, useRef } from 'react'

const Messages = ({ messages }) => {

  const messagesEndRef = useRef(null)

  const scrollToBottom = () => {
    messagesEndRef.current?.scrollIntoView({ behavior: "smooth" })
  }

  useEffect(() => {
    scrollToBottom()
  }, [messages]);

  return (
    <div>
      {messages.map(message => <Message key={message.id} {...message} />)}
      <div ref={messagesEndRef} />
    </div>
  )
}

Also made a (very basic) codesandbox if you wanna check the behaviour

Do not use findDOMNode

Class components with ref

class MyComponent extends Component {
  componentDidMount() {
    this.scrollToBottom();
  }

  componentDidUpdate() {
    this.scrollToBottom();
  }

  scrollToBottom() {
    this.el.scrollIntoView({ behavior: 'smooth' });
  }

  render() {
    return <div ref={el => { this.el = el; }} />
  }
}

Function components with hooks:

import React, { useRef, useEffect } from 'react';

const MyComponent = () => {
  const divRef = useRef(null);

  useEffect(() => {
    divRef.current.scrollIntoView({ behavior: 'smooth' });
  });

  return <div ref={divRef} />;
}

Thanks to @enlitement

we should avoid using findDOMNode, we can use refs to keep track of the components

render() {
  ...

  return (
    <div>
      <div
        className="MessageList"
        ref={(div) => {
          this.messageList = div;
        }}
      >
        { messageListContent }
      </div>
    </div>
  );
}



scrollToBottom() {
  const scrollHeight = this.messageList.scrollHeight;
  const height = this.messageList.clientHeight;
  const maxScrollTop = scrollHeight - height;
  this.messageList.scrollTop = maxScrollTop > 0 ? maxScrollTop : 0;
}

componentDidUpdate() {
  this.scrollToBottom();
}

reference:

The easiest and best way I would recommend is.

My ReactJS version: 16.12.0