This Bluetooth module ties directly into the Tx and Rx lines of the Arduino so a wireless serial connection between an Android device can be established.

An Android* phone or tablet running Android and has Bluetooth. Connect your Intel® Edison board to a Wi-Fi* network, see Get started with Intel® Edison technology.

Instructables user ASCAS shows off how to add voice control to your Arduino projects using your Android phone and Bluetooth. The Arduino gets a Bluetooth attachment and a set of code.

This project incorporates communication between an Android Phone and the Arduino using Bluetooth. The basic functionality includes taking messages from an android phone, displaying them on an LCD ...

Looking for an easy way to control that Arduino powered robot you built? We just caught wind of a pair of apps for Android that look pretty handy! WiFi Bot Control and Bluetooth Bot Control. So ...

Have you ever wanted to find an inexpensive way of communicating with your Android phone wirelessly, but without relying on the likes of Bluetooth or Google ADK? If you also happen to be one of those ...

For now, it's only possible to connect to an Arduino board through USB via an OTG cable, but joaomcgd says he plans on adding support for Bluetooth or Wi-Fi receivers in the future.

The $22 kit includes an Arduino-BLE board and 6-pin header that should be fairly broadly applicable for building Arduino-based hardware that can communicate with iOS devices.