Curves, shapes and images may undergo very large deformations while still remaining within a single semantic category, for instance an anatomical structure, human face or handwritten character. These transformations form an essential component of image understanding. A variety of spaces have been proposed as natural domains, especially finite-dimensional and infinite-dimensional manifolds and Lie groups (e.g., of diffeomorphisms). Abstract formulations have led to feasible, if intensive, computational algorithms for elastic matching based on geodesics in order to quantify differences between shapes. Possible topics also include hierarchical shape clustering, shape learning, shape synthesis and statistics on manifolds.