I use marked (GitHub).

I first import it like this:

import marked from "marked";

I then fetch my *.md file within React's componentDidMount event and store it in my component's state using marked(text) (where text is the response):

componentDidMount() {
  const readmePath = require("./Readme.md");

  fetch(readmePath)
    .then(response => {
      return response.text()
    })
    .then(text => {
      this.setState({
        markdown: marked(text)
      })
    })
}

...and finally I render it on the page using the dangerouslySetInnerHTML attribute:

render() {
  const { markdown } = this.state;

  return (
    <section>
      <article dangerouslySetInnerHTML={{__html: markdown}}></article>
    </section>
  )
}

A full working example with react-markdown:

import React, { Component } from 'react'
import ReactMarkdown from 'react-markdown'
import termsFrPath from './Terms.fr.md'

class Terms extends Component {
  constructor(props) {
    super(props)

    this.state = { terms: null }
  }

  componentWillMount() {
    fetch(termsFrPath).then((response) => response.text()).then((text) => {
      this.setState({ terms: text })
    })
  }

  render() {
    return (
      <div className="content">
        <ReactMarkdown source={this.state.terms} />
      </div>
    )
  }
}

export default Terms

You should use react-markdown instead of the accepted answer, this solution doesn't use dangerouslySetInnerHTML.

App.js

import React, { Component } from 'react';
import AppMarkdown from './App.md';
import ReactMarkdown from 'react-markdown';

class App extends Component {

  constructor() {
    super();
    this.state = { markdown: '' };
  }

  componentWillMount() {
    // Get the contents from the Markdown file and put them in the React state, so we can reference it in render() below.
    fetch(AppMarkdown).then(res => res.text()).then(text => this.setState({ markdown: text }));
  }

  render() {
    const { markdown } = this.state;
    return <ReactMarkdown source={markdown} />;
  }
}

export default App;

App.md

# React & Markdown App
* Benefits of using React... but...
* Write layout in Markdown!