1781年，Gaspard Monge定义了“很优运输”问题（即以可能的很小工作量进行质量转移），并想到将其应用于工程。1942年，Leonid Kantorovich将新生的线性规划用于Monge问题，并想到将其应用于经济。1987年，Yann Brenier利用很优运输证明了一个保持映射的度量集上新的规划定理，并想到将其应用于流体力学。每一个这样的贡献都标志着一个完整数学理论的开端，它有很多意料不到的分支。当前，研究人员从极其多样化的视角来使用和研究Monge—Kantorovich问题，这包括概率论、泛函分析、等周问题、偏微分方程乃至气象学。

Preface of the Second Edition
Preface of the First Edition
Notation
Introduction
1.Formulation of the optimal transportation problem
2.Basic questions
3.Overview of the course
Chapter 1.The Kantorovich Duality
1.1.Generalduality
1.2.Distance cost functions
1.3.Appendix： A duality argument in Cb（X × Y）
1.4.Appendix： {0，1}—valued costs and Strassen's theorem
Chapter 2.Geometry of Optimal Transportation
2.1.A duality—based proof for the quadratic cost
2.2.The real line
2.3.Alternative arguments
2.4.Generalizations to other costs
2.5.More on c—concave functions
Chapter 3.Brenier's Polar Factorization Theorem
3.1.Rearrangements and polar factorization
3.2.Historical motivations： fluid mcchanics
3.3.Proof of Brenier's polar factorization theorem
3.4.Related facts
Chapter 4.The Monge—Ampere Equation
4.1.Informal presentation
4.2.Regularity
4.3.Open problems
Chapter 5.Displacement Interpolation and Displacement Convexity
5.1.Displacement interpolation
5.2.Displacement convexity
5.3.Application： uniqueness of ground state
5.4.The Eulerian point of view
Chapter 6.Geometric and Gaussian Inequalit.ies
6.1.Brunn—Minkowski and Prekopa—Leindler inequalities
6.2.The Alesker—Dar—Milman diffeomorphism
6.3.Gaussian inequalities
6.4 Sobolev ineqiialities
Chapter 7.The Metric Side of Optimal Transportation
7.1.Monge—Kantorovich distances
7.2.Topological properties
7.3.The real line
7.4.Behavior under rescalcd convolution
7.5.An application to the Boltzmann equation
7.6.Linearization
Chapter 8.A Differential Point of View on Optimal Transportation
8.1.A differential formulation of optimal transportation
8.2.Differential calculus in （P（Rn）， W2）
8.3.Monge—Kantorovich induced dynamics
8.4.Time—discretization
8.5.Differentiability of the quadratic Wasserstein distance
8.6.Non—quadratic costs
Chapter 9.Entropy Production and Transportation Inequalities
9.1.More on optimal—transportation induced dissipative equations
9.2.Logarithmic Sobolev inequalities
9.3.Talagrandinequalitics
9.4.HWI inequalities
9.5.Nonlincar generalizations： internal energy
9.6.Nonlinear generalizat.ions： interaction energy
Chapter 10.Problems
List of Problems
Bibliography
Table of Short Statements
Index