ReactJS Module build failed: SyntaxError: Unexpected token - ReactDOM.render

I was getting the same error, and I simply needed to add a .babelrc file to the route of the project (same location as package.json) containing the following:

{
    "presets": [
        "env", 
        "react"
    ],
    "plugins": [
        "transform-class-properties"
    ]
}

Use this as your webpack config file

const webpack = require('webpack');
const path = require('path');

module.exports = {
    entry: [
        './src/app.js'
    ],
    output: {
        path: path.resolve(__dirname, './build'),
        filename: 'app.bundle.js',
    },
    module: {
        loaders: [
            {
                test: /\.html$/,
                loader: 'file-loader?name=[name].[ext]',
            },
            {
                test: /\.jsx?$/,
                exclude: /node_modules/,
                loader: 'babel-loader',
            query: {
                presets: ['es2015', 'react']
            }
        },
    ],
},
plugins: [
    new webpack.NamedModulesPlugin(),
]
};

You are missing the presets