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PrefaceConventionsIntroduction 1 Couplings and changes of variables 2 Three examples of coupling techniques 3 The founding fathers of optimal transportPart I Qualitative description of optimal transport 4 Basic properties 5 Cyclical monotonicity and Kantorovich duality 6 The Wasserstein distances 7 Displacement interpolation 8 The Monge-Mather shortening principle 9 Solution of the Monge problem I: Global approach 10 Solution of the Monge problem II: Local approach 11 The Jacobian equation 12 Smoothness 13 Qualitative picturePart II Optimal transport and Riemannian geometry 14 Ricci curvature 15 Otto calculus 16 Displacement convexity I 17 Displacement convexity II 18 Volume control 19 Density control and local regularity 20 Infinitesimal displacement convexity 21 Isoperimetric-type inequalities 22 Concentration inequalities 23 Gradient flows I 24 Gradient flows II: Qualitative properties 25 Gradient flows III: Functional inequalitiesPart III Synthetic treatment of Ricci curvature 26 Analytic and synthetic points of view 27 Convergence of metric-measure spaces 28 Stability of optimal transport 29 Weak Ricci curvature bounds I" Definition and Stability 30 Weak Ricci curvature bounds II: Geometric and analytic propertiesConclusions and open problems References List of short statements List of figures Index Some notable cost functions