  We introduce the spherical wedge billiard, a dynamical system consisting of a
particle moving along a geodesic on a closed non-Euclidean surface of a
spherical wedge. We derive the analytic form of the corresponding Poincar\'e
map and find very complex dynamics, ranging from completely chaotic to very
regular, exhibiting fractal features. Further, we show that upon changing the
billiard parameter, the fixed points of the Poincar\'e map merge in complex
ways, which has origin in the spherical aberration of the billiard mapping. We
also analyze in detail the regular regime when phase space diagram is closely
related to Talbot carpets.
