This is the first comprehensive introduction to the theory of mass transportation with its manyand sometimes unexpectedapplications. In a novel approach to the subject, the book both surveys the topic and includes a chapter of problems, making it a particularly useful graduate textbook. In 1781, Gaspard Monge defined the problem of "optimal transportation" (or the transferring of mass with the least possible amount of work), with applications to engineering in mind. In 1942, Leonid Kantorovich applied the newborn machinery of linear programming to Monge's problem, with applications to economics in mind. In 1987, Yann Brenier used optimal transportation to prove a new projection theorem on the set of measure preserving maps, with applications to fluid mechanics in mind. Each of these contributions marked the beginning of a whole mathematical theory, with many unexpected ramifications. Nowadays, the MongeKantorovich problem is used and studied by researchers from extremely diverse horizons, including probability theory, functional analysis, isoperimetry, partial differential equations, and even meteorology. Originating from a graduate course, the present volume is intended for graduate students and researchers, covering both theory and applications. Readers are only assumed to be familiar with the basics of measure theory and functional analysis. Readership Graduate students and research mathematicians interested in probability theory, functional analysis, isoperimetry, partial differential equations, and meteorology. Reviews "Villani writes with enthusiasm, and his approachable style is aided by pleasant typography. The exposition is far from rigid. ... As an introduction to an active and rapidly growing area of research, this book is greatly to be welcomed. Much of it is accessible to the novice research student possessing a solid background in real analysis, yet even experienced researchers will find it a stimulating source of novel applications, and a guide to the latest literature."  Geoffrey Burton, Bulletin of the LMS "Cedric Villani's book is a lucid and very readable documentation of the tremendous recent analytic progress in `optimal mass transportation' theory and of its diverse and unexpected applications in optimization, nonlinear PDE, geometry, and mathematical physics."  Lawrence C. Evans, University of California at Berkeley "The book is clearly written and well organized and can be warmly recommended as an introductory text to this multidisciplinary area of research, both pure and applied  the mass transportation problem."  Studia Universitatis BabesBolyaiMathematica "This is a very interesting book: it is the first comprehensive introduction to the theory of mass transportation with its many  and sometimes unexpected  applications. In a novel approach to the subject, the book both surveys the topic and includes a chapter of problems, making it a particularly useful graduate textbook."  Olaf Ninnemann for Zentralblatt MATH Table of Contents  Introduction
 The Kantorovich duality
 Geometry of optimal transportation
 Brenier's polar factorization theorem
 The MongeAmpère equation
 Displacement interpolation and displacement convexity
 Geometric and Gaussian inequalities
 The metric side of optimal transportation
 A differential point of view on optimal transportation
 Entropy production and transportation inequalities
 Problems
 Bibliography
 Table of short statements
 Index
