  Two notions of riffle shuffling on finite Coxeter groups are given: one using
Solomon's descent algebra and another using random walk on chambers of
hyperplane arrangements. These coincide for types $A$,$B$,$C$, $H_3$, and rank
two groups. Both notions have the same, simple eigenvalues. The hyperplane
definition is especially natural and satisfies a positivity property when $W$
is crystallographic and the relevant parameter is a good prime. The hyperplane
viewpoint suggests interesting connections with Lie theory and leads to a
notion of riffle shuffling for arbitrary real hyperplane arrangements and
oriented matroids. Connections with Cellini's descent algebra are given.
