Official iOS App - Pair, program and flash programs to the micro:bit via Bluetooth.
Official Swift Playgrounds - (Source Code) Swift Playgrounds is an app for the iPad that helps teach people to code in the Swift language using interactive 'books'.
micro:bit Blue - (Source Code) Android app that contains a series demos for interacting with the micro:bit using Bluetooth.
Bitty Software Apps - Diverse collection of Android and iOS apps, going from data logging to audio pranks, you'll certainly find something of interest.
Insight Mr Bit - (iOS) Create simple programs in plain English to control the BBC micro:bit to do lots of useful things.
micro:bit Xamarin - Open source Android app that communicates with the micro:bit over BLE and gets sensor data. A good example of using Xamarin (a cross platform mobile framework) with the micro:bit.
bitty blue - iOS and Android app that provides a collection of fun things to do with a BBC micro:bit (or compatible device) and Bluetooth.
micro:bit logger - Android app that enables users to log data from the BLE services and export it to a file.
Kitronik Move - Android app that presents a D-Pad interface to control a micro:bit over Bluetooth LE.
nRF Connect - A generic tool for Android that allows you to scan, advertise and explore BLE devices. It supports the micro:bit by including information on the micro:bit services, custom macros and more.
Tickle - iOS app to program a large selection of devices, including the micro:bit, connecting them all together, so that they can interact with each other.
micro:bit Slither - Multiplayer snake game on Android, controlled with micro:bits via Bluetooth.
Micro:bit Explorer - A Swift Playground Book that introduces computer fundamentals in a visual way, it allows you to enter machine code or assembly and see how it executes and how the micro:bit registers are affected.
BirdBlox - A tablet-based programming option for the Hummingbird, a robotics board kit for the micro:bit.