The overview includes only games which are completely free, with a permissive license for both the code and their content. I could not include here games like Urban Terror or Warsow, since their assets are not free, nor Steam games.

A very beautiful game, 0 A.D. is a 3D medieval strategy game with different nations, incorporating elements from various other games pertaining to the genre, like Age of Empires. Still in alpha stage (and has been for a long time), 0 A.D. is playable in both single-player and online. It has great graphics, specific units and buildings for each faction, and a selection of carefully designed maps.
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Red Eclipse is a shooter game set in futuristic environments, including many modes, game mutators, and trickjumps like dashing or impulse boosts. Just like Sauerbraten below, it’s based on the Cube 2 engine.
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Hedgewars is a Worms-like game which preserves the original feeling, but brings some really neat features like user registration and online playing, new maps, game modes, sounds and graphics. For the fans of the Worms series, this is definitely a really good, free alternative.
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This a the very popular and very addictive turn-based game. The Battle for Wesnoth takes place in fantasy universes, with 7 different factions available, and battles taking place over hexagonal tiles, with different unit abilities, traits and terrain defenses. Although it has some simple rules, the gameplay can be very dynamic and complex, with matches ranging from minutes up to hours.
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OpenArena takes the ioQuake3 engine and replaces all the Quake 3 assets with free ones. The result is a completely free Quake 3 clone, with its own maps, player and weapon models, sounds, and a good online playing experience. OpenArena brings some really nice game modes, including mutators (like Rocket-only mutator or Instantgib mutator).
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This is a Mario clone with Tux as the protagonist. SuperTux includes several missions with lots of levels, all the monsters and techniques specific to the genre.
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This is a real-time strategy game focused on research of new technologies, with a very ample tech-tree due to the available possibilities.
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This is based on the Cube engine, it is a first-person shooter game somewhat different than the casual players. There are lots of maps, weapons and especially game modes, like the popular CTF ones (including Keep the Flag or Hunt the Flag) or the One Shot, One Kill one. Trickjumps are also possible, and they are a bit different in AC in that you can also use the so-called “rifle jumps” to propel yourself higher.
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Based on the VDrift and OGRE engines, Stunt Rally is a 3D-intensive racing game with beautiful scenery and several interesting tracks.
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This is a ball-driving game, with the goal of running the ball all over various maps and collecting all the gold without falling off. Neverball can be real fun to play, however it requires great skills and tricks in order to master it, especially on high-difficulty levels.
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Supertuxkart is a popular racing game with a funny twist. You can choose from various missions and complete different tasks to unlock more tracks. It’s quite a nice compared to all the more “serious” racing games.
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Based on the Cube 2 engine, Sauerbraten is a shooter that provides a bunch of game modes, ranging from DM to the CTF family or Instagib.
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A few more to include here: Alien Arena, Tremulous or World of Padman (all of which are first-person shooters with a Sci-Fi flavour).

Although not completely open-source due to a less permissive assets license, worth mentioning are also Xonotic, Urban Terror and Warsow.

Have some more? Please share them in the comments below.