A collection of hyperplanes in Euclidian space divides space into chambers and faces. Bidigare, Hanlon, and Rockmore introduced natural random walks on the chambers. This includes classical examples (Tsetlin library card shuffling) and many new examples (territory exchange, walks on a-cyclic orientations). There is a fairly complete theory, but many questions remain. This is joint work with Christos Athanasiadis.