ioquake3 (or ioq3 for short) builds upon id Software's Quake 3 source code release. It cleans it up, fixes bugs, and adds features. Its goal is to be the Quake 3 distribution upon which people base their games and projects. Its design goals include optimizing the engine for playing Quake 3: Arena, Team Arena, and all popular mods. This distribution of the engine has been ported to many new platforms. While it doesn't have PunkBuster (and never will), it does provide more security for servers and clients by way of various bugfixes which aren't in id's client.
Bitfighter is a game that feels a little like Asteroids on steroids. It's a multiplayer space-based combat and strategy game, featuring customized ships, multiple game modes, cool retro graphics, and smooth network play. It also includes a full featured level editor, Lua scriptable bots, and other cool features. It has a very active development team, so the game keeps improving.
Arx Libertatis is a cross-platform, open source port of Arx Fatalis, a 2002 first-person role-playing game developed by Arkane Studios. Arx Fatalis features crafting, melee and ranged combat, and a unique casting system where the player draws runes in real time to effect the desired spell. The Arx Libertatis source code is based on the publicly released Arx Fatalis sources. This does however not include the game data, so you need to obtain a copy of the original Arx Fatalis or its demo.