Abstract: Here Shape space consists of either: sets of ordered points on Rⁿ (landmark space), submanifolds of Rⁿ, even certain currents in Rⁿ. The group of diffeomorphisms with compact support acts transitively on shape space, and we induce Riemannian metrics on shape space induced from right invariant Sobolev metrics in the diffeomorphism groups. This offers interesting analytical difficulties. The resulting geodesic equations are well posed. Curvature can be computed in two ways: directly, or via O'Neill's formula from Arnold's curvature on the diffeomorphism group.