It is understood, already since D'Arcy Thompson's seminal studies on biological forms, that deformations are natural objects for addressing issues related to shape comparison. Stemming from Grenander's theory of deformable templates, these ideas are formalized by the study of groups of diffeomorphisms, and on the way they can act on shapes and images. This talk will provide basic concepts relative to these groups, and how theory can be used to lead to feasible algorithms. We will then review several possible actions, on different mathematical objects that can be used to model shapes, and discuss some applications.