After a brief introduction to non-Euclidean 2D geometry using fleece surfaces (tecido polar), volunteers will put on a VR headset to try their hand at billiards in 3D spherical, Euclidean and hyperbolic spaces, with the rest of the audience following along on the lecture hall’s main screen. Even experienced geometers may find some surprising optical effects, which we’ll explain using the fleece surfaces. The VR Curved-Space Billiards game will remain available in the lecture hall for an hour or two after the talk to let as many people as possible try non-Euclidean billiards for themselves.