We have not yet had an official release. Nonetheless, you can still get icculus.org/quake3 packaged for a few Linux distributions or build it yourself from Subversion.
The Quake 3 engine is open source, this does not mean that Quake 3 Arena the game is free. You must purchase the game to use the data!
If you're lucky your distribution or someone else using it may have already done all of the work for you.
Gentoo users can just emerge quake3 to get a recent snapshot of our Subversion repository.
Ludwig Nussel works on this project and also provides RPMs for SUSE Linux Enterprise Server. Note that these are just the binaries, no scripts or KDE/Gnome launchers to start them. Ludwig also mentions that the packages should work on any glibc 2.3 system.
There are i686 binaries built on Ubuntu made by Pascal de Bruijn. Untar this into /opt/ or whereever you keep weird isolated installs. There is a decent chance these will work on most distributions, these are not .debs.
This part assumes Windows users are using Cygwin. If you aren't, TortoiseSVN is a good client.
Everyone else can compile it themselves
In a terminal, type make in the quake 3 directory you've downloaded with subversion, then install the latest id patch for OS X (at the bottom of this page), then copy the resulting darwinquake3 binary in code/unix/releaseppc/ to /Applications/Quake3/Quake3.app/Contents/MacOS/Quake3. You also need an /libSDL-1.2.0.dylib in that .app's MacOS directory. You'll need to copy your pak0.pk3 from your Quake 3 cd-rom's baseq3 directory to your /Applications/Quake3/baseq3/ directory. Ditto for the Team Arena mission pack, though you'd be changing baseq3 in the above example to missionpack.
After you've done all of that mess, you can double click on the .app file, or cd to /Applications/Quake3/in a terminal and type Quake3.app/Contents/MacOS/Quake3, and hit return.
Building on Windows is slightly complicated. You can either use Microsoft Visual C++ or MinGW. MinGW works better currently; both methods are described in the README you already are supposed to have read.
The id software patches are still needed before installing this project because they contain updated pk3 files. We mirror them for your convienience: