This book, "Topics in Optimal Transportation" by Cédric Villani, is a comprehensive introduction to the theory of mass transportation, covering its historical development and various applications. The book begins with the problem defined by Gaspard Monge in 1781, which was later applied to economics by Leonid Kantorovich in 1942 and to fluid mechanics by Yann Brenier in 1987. These contributions have led to a broad mathematical theory with applications in probability theory, functional analysis, isoperimetry, partial differential equations, and meteorology. The book is structured to survey the topic and includes a chapter of problems, making it suitable as a graduate textbook. It is not intended to be exhaustive but rather an introduction, complemented by recent reference texts such as those by S. T. Rachev and L. Rüschendorf, Evans, L. Ambrosio, and J. Urbas. The volume covers displacement interpolation, functional inequalities, and the differential viewpoint of Otto, while also providing intuition and recent research without proofs. It is designed for graduate students and researchers with a background in measure theory and functional analysis.This book, "Topics in Optimal Transportation" by Cédric Villani, is a comprehensive introduction to the theory of mass transportation, covering its historical development and various applications. The book begins with the problem defined by Gaspard Monge in 1781, which was later applied to economics by Leonid Kantorovich in 1942 and to fluid mechanics by Yann Brenier in 1987. These contributions have led to a broad mathematical theory with applications in probability theory, functional analysis, isoperimetry, partial differential equations, and meteorology. The book is structured to survey the topic and includes a chapter of problems, making it suitable as a graduate textbook. It is not intended to be exhaustive but rather an introduction, complemented by recent reference texts such as those by S. T. Rachev and L. Rüschendorf, Evans, L. Ambrosio, and J. Urbas. The volume covers displacement interpolation, functional inequalities, and the differential viewpoint of Otto, while also providing intuition and recent research without proofs. It is designed for graduate students and researchers with a background in measure theory and functional analysis.