Eight fine Quake III Arena mods were published on Desura.

Most of them (six!) are total conversions.

Main link to source: www.desura.com

Total conversions: Q3 Rally, Urban Terror, Bid for Power, Combat, True Combat, Navy SEALs: Covert Operations

Partial conversions extending the core gameplay: David Hunter and Excessive +

You can play these mods via retail ("play") or Steam ("play through Steam") if you have a Steam copy of Quake III Arena. Also note that Quake III Arena and consequently its mods are mostly multiplayer-oriented, so you'll probably have to grab your buddies to play these mods (as zero to few servers are up at the moment).

Excessive +

Got tired of standard CPMA/QuakeLive ? Excessive Plus is for you, the fastest, most explosive 3D Action/Shooter (FPS) active mod in the world. It works with Quake III, IoQuake3, OpenArena and Xreal. What's possible on Quake3 excessiveplus ? Shortly, you can do anything with excessiveplus ! So join it, and enjoy it ;] "E+ is modification where all weapons and movement, are super-powerful and in every way excessive."

Skullhead T35T movie - Desura

Skullhead T35T movie - YouTube

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All other available mods:

Mods install to the main game directory. You can also set your own mod launch commands on Desura, a short User Launch Options tutorial is here.