Optimal Transportation and Applications : Lectures given at the C.I.M.E. Summer School held in Martina Franca Italy September 2 8 2001 Leading researchers in the field of Optimal Transportation with different views and perspectives contribute to this Summer School volume: Monge Ampère and Monge Kantorovich theory shape optimization and mass transportation are linked among others to applications in fluid mechanics granular material physics and statistical mechanics emphasizing the attractiveness of the subject from both a theoretical and applied point of view. The volume is designed to become a guide to researchers willing to enter into this challenging and useful theory