Free and open source games, some of which are also available on Steam (if they're also on Steam, I'll mark them with a *).
- Shooters
https://xonotic.org/ - Quake-like Arena shooter
*https://redeclipse.net/ - Another quake-like arena shooter
- Strategy
https://mindustrygame.github.io/
*https://wz2100.net/ - vehicle RTS with customizable units
*https://www.wesnoth.org/ - Turn/grid based strategy game.
http://www.openra.net/ - FOSS reimplimentation of RedAlert
http://www.freeciv.org/ - Turn based Grand Strategy
https://github.com/yairm210/UnCiv - "
https://play0ad.com/ - unit/civ based RTSĀ
https://zero-k.info/
https://www.freeorion.org/ - Space based grand strategy/4X
- Platformers
https://www.srb2.org/ - 2.5D Sonic the Hedgehog game
http://www.cavestory.org/ - Side scrolling platformer/shmup (Note: The Steam version is not free, it was basically stolen from the OG dev)
https://www.supertux.org/ - 2d platformer
- Racing
https://supertuxkart.net/Main_Page
https://sourceforge.net/projects/torcs/
- RPGs
https://www.themanaworld.org/ - FOSS MMORPG
http://www.cataclysmdda.org/ - Turn based survival game
https://sulisgame.com/
https://oolite.space/ - Custom FOSS build of ELITE
https://pioneerspacesim.net/ - Space adventure and exploration
http://www.freedroid.org/ - iso-rpg
*https://naev.org/ - Space sim
*https://endless-sky.github.io/ - Space sim (Like Escape Velocity, if you remember that)
- Misc
https://www.luanti.org/ - Voxel game engine, but you can install games from within it, including Minetest and VoxelLibre
https://veloren.net/ - Voxel adventure/dungeon crawler
https://www.openttd.org/ - FOSS Transport Tycoon