  This article provides an overview of various notions of shape spaces,
including the space of parametrized and unparametrized curves, the space of
immersions, the diffeomorphism group and the space of Riemannian metrics. We
discuss the Riemannian metrics that can be defined thereon, and what is known
about the properties of these metrics. We put particular emphasis on the
induced geodesic distance, the geodesic equation and its well-posedness,
geodesic and metric completeness and properties of the curvature.
