Abstract: The metric completeness, geodesic completeness, and geodesic convexity of shape spaces, such as diffeomorphism groups, spaces of immersed curves, and spaces of immersed surfaces, have been studied intensively over the last twenty years. As in finite dimensions, geodesic completeness is a natural starting point for studying the existence of periodic geodesics. In infinite dimensions, however, the existence of periodic geodesics remains largely open, apart from a few special cases. We present existence results for periodic geodesics on half-Lie groups equipped with strong right-invariant Riemannian metrics under suitable regularity and completeness assumptions. Examples include Sobolev diffeomorphism groups. We also discuss analogous results for more general strong Riemannian Hilbert manifolds, where additional compactness assumptions, such as Palais--Smale type conditions, are required. This is joint work in progress with Martin Bauer.