ADX

ADX is a sound format used in Persona 3 and Persona 4. It was developed by CRI Middleware and first used in 1996 for Sega Saturn titles. ADX can be found in CVM, ACX, and AFS archives. Usage ranges from non-looping sound effects and voices to looping background music. For proper conversion and playback in-game, do not exceed 24000 Hz on PS2, or 22050 Hz on PSP.

adxencd.exe
Commandline tool ADX Audio Decompression Tool v4.61 can be used to encode or decode ADX files. Using samples, you can even set loop points for a BGM track. Download adxencd.exe here.

DKZStudio
DKZ can batch-convert WAV files to ADX, and also play back ADX sounds using its built-in media player. The latest available version is 0.92 Beta. It was originally released in Spanish, but has support for Language packs in DSL format. An English language DSL is included in the download. However, you cannot set loop points using DKZ as the only option is to loop a sound from start to end.

PES Sound File Converter
PES v1.8 can also batch convert WAV files to ADX and back. You can download PES here.

XMPlay
XMPlay is a third party music player with a simple UI which allows high quality .ADX playback and encoding. You are able to encode .ADX files into FLAC, MP3, OGG, and more. However, converting to .ADX is not possible using XMPlay. Download vanilla (No plugins) XMPlay here, or download the all-in one folder (Contains plugins which allow .ADX playback and encoding) here.

VLC Media Player
VLC media player can play ADX files.