An algorithm is presented for generating pseudorandom variates distributed uniformly over an arbitrary convex polytope in a Euclidean space of arbitrary dimension. Many commonly used methods for generating uniform variates require that the polytope be expressed as the solution set of a system of linear inequalities; the algorithm presented here requires instead that the polytope be presented in terms of a finite generating set, typically the set of its vertices. Included in the algorithm are procedures for identifying all faces of the polytope and for decomposing the polytope into simplices.