Shape spaces can be endowed with the structure of Rieman- nian 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 paper we summarize results contained in [1], which deals with the computation of curvature for landmark shape spaces. Implications on both the qualitative dynamics of geodesies and the statistical analysis on shape manifolds are also discussed.