Try to think of the tags as function calls (see docs). Then the first one becomes:

{[1,2,3].map(function (n) {
  return React.DOM.p(...);
})}

And the second one:

{[1,2,3].map(function (n) {
  return (
    React.DOM.h3(...)
    React.DOM.p(...)
  )
})}

It should now be clear that the second snippet doesn't really make sense (you can't return more than one value in JS). You have to either wrap it in another element (most likely what you'd want, that way you can also provide a valid key property), or you can use something like this:

{[1,2,3].map(function (n) {
  return ([
    React.DOM.h3(...),
    React.DOM.p(...)
  ]);
})}

With JSX sugar:

{[1,2,3].map(function (n) {
  return ([
    <h3></h3>, // note the comma
    <p></p>
  ]);
})}

You don't need to flatten the resulting array, React will do that for you. See the following fiddle http://jsfiddle.net/mEB2V/1/. Again: Wrapping the two elements into a div/section will most likely be better long term.

It seems the old answer about returning an array no longer applies (maybe since React ~0.9, as @dogmatic69 wrote in a comment).

The docs say you need to return a single node:

Maximum Number of JSX Root Nodes

Currently, in a component's render, you can only return one node; if you have, say, a list of divs to return, you must wrap your components within a div, span or any other component.

Don't forget that JSX compiles into regular JS; returning two functions doesn't really make syntactic sense. Likewise, don't put more than one child in a ternary.

In many cases you can simply wrap things in a <div> or a <span>.

In my case, I wanted to return multiple <tr>s. I wrapped them in a <tbody> – a table is allowed to have multiple bodies.

EDIT: As of React 16.0, returning an array is apparently allowed again, as long as each element has a key: https://facebook.github.io/react/blog/2017/09/26/react-v16.0.html#new-render-return-types-fragments-and-strings

EDIT: React 16.2 lets you surround a list of elements with <Fragment>…</Fragment> or even <>…</>, if you prefer that to an array: https://blog.jmes.tech/react-fragment-and-semantic-html/

From React v16.0.0 onwards, it is possible to Return multiple elements by wrapping them within an Array

render() {
  return (
    {[1,2,3].map(function (n) {
      return [
        <h3>Item {n}</h3>.
        <p>Description {n}</p>
      ]
    }}
  );
}

Also from React v16.2.0, a new feature called React Fragments is introduced which you can use to wrap multiple elements

render() {
  return (
    {[1,2,3].map(function (n, index) {
      return (
        <React.Fragment key={index}>
            <h3>Item {n}</h3>
            <p>Description {n}</p>
        </React.Fragment>
      )
    }}
  );
}

As per the documentation:

A common pattern in React is for a component to return multiple elements. Fragments let you group a list of children without adding extra nodes to the DOM.

Fragments declared with the explicit syntax may have keys. A use case for this is mapping a collection to an array of fragments — for example, to create a description list:

function Glossary(props) {
  return (
    <dl>
      {props.items.map(item => (
        // Without the `key`, React will fire a key warning
        <React.Fragment key={item.id}>
          <dt>{item.term}</dt>
          <dd>{item.description}</dd>
        </React.Fragment>
      ))}
    </dl>
  );
}

key is the only attribute that can be passed to Fragment. In the future, we may add support for additional attributes, such as event handlers.

Also, you might want to return several list items in some helper function inside a React component. Just return an array of html nodes with the key attribute:

import React, { Component } from 'react'

class YourComponent extends Component {
  // ...

  render() {
    return (
      <ul>
        {this.renderListItems()}
      </ul>
    )
  }      

  renderListItems() {
    return [
      <li key={1}><a href="#">Link1</a></li>,
      <li key={2}><a href="#">Link2</a></li>,
      <li key={3} className="active">Active item</li>,
    ]
  }
}

You can use createFragment here.

https://facebook.github.io/react/docs/create-fragment.html

import createFragment from 'react-addons-create-fragment';
...
{[1,2,3].map((n) => createFragment({
    h: <h3>...</h3>,
    p: <p>...</p>
  })
)}

(using ES6 and JSX syntax here)

you first have to add the react-addons-create-fragment package :

npm install --save react-addons-create-fragment

Advantage over Jan Olaf Krems's solution : react does not complain about the missing key

Updated

Use React Fragment. It's simple. Link to fragment documentation.

render() {
  return (
    <>
    {[1,2,3].map((value) => <div>{value}</div>)}
    </>
  );
}

Old answer - obsolete

With React > 16 you can use react-composite.

import { Composite } from 'react-composite';

// ...

{[1,2,3].map((n) => (
  <Composite>
    <h2>Title {n}</h2>
    <p>Description {n}</p>
  </Composite>
))};

Of course, react-composite has to be installed.

npm install react-composite --save