Table of Contents

Introduction.- 1 An overview on Optimal Transportation.- 2 The Monge-Ampère Equation.- 3 Sobolev regularity of solutions to the Monge-Ampère equation.- 4 Second order stability for the Monge-Ampère equation and applications.- 5 The semigeostrophic equations.- 6 Partial regularity of optimal transport maps.- A. Properties of convex functions.- B. A proof of John Lemma.- Bibliography.