In this paper, we study the so-called shake-and-bake algorithms which, as far as we know, provide the only practical way of generating (asymptotically) uniform points on the boundary as of a fully-dimensional polyiope S. The fact that the points are uniformly distributed means that, in the long run, all subsets of as of equal size will be visited equally often, and subsets with positive measure will be visited with probability one. Part of the material that we present in this paper may be found in more detail in the technical reports of Boender et al. ( l 988a, b ). For the so-called hit-and-run algorithms, which generate asymptotically uniform points on the interior of S, we refer to Smith ( 1984) and Berbee et al. ( 1987).