Shape analysis aims at quantifying variability in ‘shapes’ (e.g. images, densities, landmarks, etc.) through the analysis of deviations from a generic ‘shape’ called a template. Deviations from the template are assumed to be induced by the action of topology-preserving deformations. In this talk, we discuss an extension of this framework, called Metamorphoses, that also accounts for topology changes. We begin by reviewing a standard topology-preserving model called LDDMM. We then define the Metamorphoses model and its extension beyond ‘standard’ spaces. Finally, we illustrate these models with a few image-matching examples.