Japan. J. Math. 4, 27–45 (2009) DOI: 10.1007/s11537-009-0855-7 Ricci curvature and measures ⋆ Jean-Pierre Bourguignon Received: 8 December 2008 / Revised: 22 February 2009 / Accepted: 2 March 2009 Published online: 28 March 2009 c  The Mathematical Society of Japan and Springer 2009 Communicated by: Toshiyuki Kobayashi Abstract. In the last thirty years three a priori very different fields of mathematics, optimal transport theory, Riemannian geometry and probability theory, have come together in a remark- able way, leading to a very substantial improvement of our understanding of what may look like a very specific question, namely the analysis of spaces whose Ricci curvature admits a lower bound. The purpose of these lectures is, starting from the classical context, to present the basics of the three fields that lead to an interesting generalisation of the concepts, and to highlight some of the most striking new developments. Keywords and phrases: Ricci curvature, geometry of spaces of measures, Bakry– ´ Emery esti- mate, optimal transport theory, lower bounds on curvature, Wasserstein distances, metric mea- sured spaces, Gromov–Hausdorff topology, entropy functionals Mathematics Subject Classification (2000): 58J65, 53C21, 49J20, 28D05, 90B06, 60E15 In the last thirty years three a priori very different fields of mathematics, opti- mal transport theory, Riemannian geometry and probability theory, have come together in a remarkable way, leading to a very substantial improvement of our understanding of what may look like a very specific question, namely the anal- ysis of spaces whose Ricci curvature admits a lower bound. As it has repeatedly happened in the history of mathematics, this could only be achieved by consid- ering the question in a much wider context than the one initially envisioned. ⋆ This article is based on the 5th Takagi Lectures that the author delivered at the University of Tokyo on October 4 and 5, 2008. J.-P. BOURGUIGNON Centre national de la recherche scientifique–Institut des Hautes ´ Etudes Scientifiques, 35, route de Chartres F-91440 Bures-sur-Yvette, France (e-mail: jpb@ihes.fr)