The problem of sliding at a nonplanar grain boundary is considered in detail. The stress field, and sliding displacement and velocity can be calculated at a boundary with a shape which is periodic in the sliding direction (a wavy or stepped grain boundary): a) when deformation within the crystals which meet at the boundary is purely elastic, b) when diffusional flow of matter from point to point on the boundary is permitted. The results give solutions to the following problems. 1) How much sliding occurs in a polycrystal when neither diffusive flow nor dislocation motion is possible? 2) What is the sliding rate at a wavy or stepped grain boundary when diffusional flow of matter occurs? 3) What is the rate of diffusional creep in a polycrystal in which grain boundaries slide? 4) How is this creep rate affected by grain shape, and grain boundary migration? 5) How does an array of discrete particles influence the sliding rate at a grain boundary and the diffusional creep rate of a polycrystal? The results are compared with published solutions to some of these problems.