Diffusion geometry, scale-space analysis, and study of heat propagation on manifolds have recently become a popular tool in data analysis in a variety of applications. In this talk, we will explore the applications of diffusion geometry to the problems of non-rigid shape representation, comparison, and retrieval. We will show that diffusion processes allow defining both local and global geometric structures. Local shape descriptors based on diffusion kernels allow representing shapes as collections of geometric "words" and "expressions" and approaching shape similarity as problems in text search and matching. Global structures are diffusion metrics, insensitive to shape deformations and topological changes. Representing shapes as metric spaces endowed with diffusion distances, we can pose the problem of shape similarity as a comparison of metric spaces using the Gromov-Hausdorff distance. As examples of applications we will show large-scale shape retrieval, correspondence computation, and detection of intrinsic symmetries in non-rigid shape.