Elastic capsules occur in nature in the form of cells and vesicles and are manufactured for biomedical applications. They are widely modeled but there are few analytical results. In this talk, complex variable techniques are used to derive semi-analytic solutions for the steady-state response and time- dependent evolution of elastic capsules in 2D Stokes flow. The analysis is complemented by spectrally accurate numerical simulations of the time- dependent evolution. We provide compelling evidence of the formation of finite-time cusp singularities, of which there are few rigorous examples in interfacial Stokes flow, with none involving elastic interfaces. We discuss the relevance of our findings to 3D capsules. This is joint work with Michael Booty and Michael Higley.