Shapes play a role in many application areas, e.g. the statistical distribution of organ shapes is examined in computational anatomy in order to detect illnesses from deviations (e.g. shape changes of the brain). For such applications one needs to provide the set of all shapes with an additional structure. This is typically a Hilbert manifold structure; some shape spaces also possess a Lie group structure. We will present so-called Riemannian and Hamiltonian approaches to shape spaces, the analysis of corresponding variational problems and PDEs as well as the numerical analysis for their numeric treatment. Some background in PDEs is required (be it PDEs or numerics for PDEs or geometric analysis); background in differential geometry may be helpful, but is not required.