HyperRogue

You are a lone adventurer in a strange, non-Euclidean world. Gather as much treasure as you can before the nasty monsters get you. Explore about 60 different lands, each with its own unique treasures, enemies, and terrain obstacles. Your quest is to find the legendary treasure, the Orbs of Yendor. Collect one of them to win! Or just ignore your quest and collect smaller treasures.
The twist is the unique, unusual geometry of the world: it is one of just few games which takes place on the hyperbolic plane. Witness a grid composed of hexagons and heptagons, straight lines which seem to be parallel, but then they diverge and never cross, triangles whose angles add up to less than 180 degrees, how extremely unlikely is it to reach the same place twice due to practically infinite size of the world (centillions of locations in just 700 steps from the starting point), and how the world seems to be rotated when you do return. All this matters for the gameplay. The game is inspired by the roguetime genre, works of M. C. Escher, and by puzzle games such as Deadly Rooms of Death.
HyperRogue is often updated! Other than the original 2D hyperbolic puzzle roguetime game, HyperRogue includes experimental modes, allowing one to change many aspects of the game for your non-Euclidean experimentation, such as racing mode, real-time "shmup" mode, 3D geometries including Thurston geometries, SteamVR virtual reality. If you want to play HyperRogue as a non-Euclidean first person shooter, it is possible too!