I am using React-Native 0.40 and below code works for me both on IOS and Android.

(Android-only) Sets the elevation of a view, using Android's underlying elevation API. This adds a drop shadow to the item and affects z-order for overlapping views. Only supported on Android 5.0+, has no effect on earlier versions.

 class MainApp extends Component {
  render() {
    return (
      <View style={styles.container}>

        <View elevation={5} style={styles.buttonContainer}>
          <Text style={styles.textStyle}>Shadow Applied</Text>
        </View>
      </View>
    );
  }
}

const styles = StyleSheet.create({
  container: {
    flex: 1,
    justifyContent: 'center',
    alignItems: 'center',
    backgroundColor: '#FFFFFF'
  },
  textStyle: {
    color: '#FFFFFF'
  },
  buttonContainer: {
    backgroundColor: '#2E9298',
    borderRadius: 10,
    padding: 10,
    shadowColor: '#000000',
    shadowOffset: {
      width: 0,
      height: 3
    },
    shadowRadius: 5,
    shadowOpacity: 1.0
  }
})

Tested on iPhone.

Edit

Comment from @ James. Thanks.

Note: For those on android, the backgroundColor is critical. I was using View as a container for another element and couldn't get a shadow until I specified a background color.

Use elevation to implement shadows on RN Android. Added elevation prop #27

<View elevation={5}> </View>

It appears to be a bug in React native that shadowOpacity is set to type CGFloat instead of float according to CALayer doc. use iPhone 5 simulator before it's fixed. (CGFloat is float in older devices. )

The React Native issue which is tracking this is:

https://github.com/facebook/react-native/issues/449