Although a variety of algorithms have been developed to synthesize new textures on regular 2D grids, this is seldom the end goal for applications. More often, what is needed is a texture that fits a given surface. In this chapter we will look at methods for performing texture synthesis on surfaces. More specifically, we will assume that a surface description and a texture exemplar has been provided, and that we wish to create a texture on that mesh that is based on the exemplar. Although many of the algorithms are described in terms of polygonal meshes, most of the methods easily extend to other kinds of surfaces (e.g. subdivision surfaces, tensor product patches).