Introduction 1
1 Couplings and changes of variables 5
2 Three examples of coupiing techniques 21
3 The founding fathers of optimal transport 29
Part Ⅰ Qualitative description of optimal transport 39
4 Basic properties 43
5 Cyclical monotonicity and Kantorovich duality 51
6 The Wasserstein distances 93
7 Displacement interpolation 113
8 The Monge-Mather shortening principle 163
9 Solution of the Monge problem Ⅰ：Global approach 205
10 Solution of the Monge problem Ⅱ：Local approach 215
11 The Jacobian equation 273
12 Smoothness 281
13 Qualitative picture 333
Part Ⅱ Optimal transport and Riemannian geometry 353
14 Ricci curvature 357
15 Otto calculus 421
16 Displacement convexity Ⅰ 435
17 Displacement convexity Ⅱ 449
18 Volume control 493
19 Density control and local regularity 505
20 Infinitesimal displacement convexity 525
21 Isoperimetric-type inequalities 545
22 Concentration inequalities 567
23 Gradient flows Ⅰ 629
24 Gradient flows Ⅱ：Qualitative properties 693
25 Gradient flows Ⅲ：Functional inequalities 719
Part Ⅲ Synthetic treatment of Ricci curvature 731
26 Analytic and synthetic points of view 735
27 Convergence of metric-measure spaces 743
28 Stability of optimal transport 773
29 Weak Ricci curvature bounds Ⅰ：Definition and Stability 795
30 Weak Ricci curvature bounds Ⅱ：Geometric and analytic properties 847
Conclusions and open problems 903
References 915
List of short statements 957
List of figures 965
Index 967
Some notable cost functions 971