Registration (for curves, surfaces or images) can be seen as a representation of a shape dataset as a subset of the diffeomorphism group, each shape in the dataset being associated with an optimal deformation of a template. We will in particular focus on methods that compute this deformation by optimizing a distance over diffeomorphisms under the constraint of mapping a template to a target, and such that the minimized distance is in turn a metric in shape space. These “metric registration� methods include a large spectrum of algorithms deriving from the “large deformation diffeomorphic metric mapping� (LDDMM) framework, and “metamorphosis� in which non-diffeomorphic shape variations are also allowed. We will describe mathematical foundations and algorithms, and the various forms that these methods may take in relation to applications.