The NEM project has two official mobile apps, which are currently in beta. The current Android version is 1.0.49, and the iOS version is 2.0. The apps do not yet allow you to use the NEM Network however, and instead they are for testing on the testnet only.
The current version of the app is fairly full featured, with one major caveat being that it can only connect to the test network, and can send no real XEM. The app allows you to generate addresses, and use them to send/receive transactions. There is also a messaging client within the app that lets you use NEM's Messaging feature in either encrypted or unencrypted forms.
In addition to messaging, the app allows you to keep an address book that allows you to associate contacts with NEM addresses. This can be used to send message to a name, rather than a meaningless string of letters. You can also scan addresses into the address book using the QR code reader.
The apps can be downloaded from the forum at: https://forum.nem.io/t/nem-mobile-app-open-beta-and-10-000-xem-bug-bounty/1825