The problem of Optimal Mass Transport, due to Monge,is to transfer a mass from one location to another trying to minimize a total cost. The case where the cost is the square of the distance on the Euclidean space was solved by Brenier almost 20 yeras ago. After this work several people showed the existence of Optimal Transport for more general costs on Euclidean spaces (among them Evans, Gangbo, McCann, Ambrosio, Pratelli, Cafarelli, Trudinger, Wang), for Riemannian distances or Lagrangian costs on compact manifolds (McCann, Bernard, Buffoni) and on certain non-compact manifolds (McCann, Feldmann). An excellent reference on the subject is Cedric Villani’s book Topics in Mass Transprtation (or even better the lecture notes for his course in Saint-Flour in Summer 2005). In a recent work with Alessio Figalli, modifying some of the existing techniques, we were able to extend the results the case where the cost is obtained from a smooth Lagrangian strictly convex in the speed. In this lecture, for a general non-specialist audience, we will explain the problem, and sketch some aspects of the proof.