Shape spaces can be endowed with the structure of Riemannian manifolds; this allows one to compute, for example, Euler-Lagrange equations and geodesic distance for such spaces. Until very recently little was known about the actual geometry of shape manifolds; in this talk we summarize results contained in my recent doctoral dissertation, which deals with the computation of curvature for "Landmarks Shape Spaces." Implications on both the qualitative dynamics of geodesics and the statistical analysis on shape manifolds are also discussed.