This text presents a comprehensive overview of statistical optimal transport, a field that combines optimal transport theory with statistical methods. The book is structured into seven main chapters and an appendix, covering foundational concepts, estimation techniques, transport maps, entropic optimal transport, Wasserstein gradient flows, and the metric geometry of the Wasserstein space. It also includes a preface, acknowledgments, and a guide on how to read the book. The book begins with an introduction to optimal transport, starting with the classical problem posed by Monge in 1781, which involves transporting sand from one location to another with minimal cost. This problem is abstracted into a problem involving probability distributions, leading to the formulation of the Kantorovich problem, which allows for more general cost functions and couplings between probability measures. The Wasserstein distance, a key concept in optimal transport, is introduced as a metric that quantifies the distance between probability distributions in a way that respects their geometry. The text then explores various aspects of optimal transport, including the estimation of Wasserstein distances, the construction of transport maps, and the use of entropic regularization. It discusses the geometric properties of the Wasserstein space, including curvature, geodesics, and gradient flows, and their applications in statistics and machine learning. The book also addresses the challenges of estimating Wasserstein distances when data are coupled, a common scenario in statistical modeling. The authors emphasize the importance of optimal transport in modern statistics and machine learning, highlighting its ability to provide a meaningful notion of distance between probability measures, which is crucial for tasks such as clustering, regression, and generative modeling. The text also discusses the computational aspects of optimal transport, including the use of the Sinkhorn algorithm and computational optimal transport, which has become a powerful tool in machine learning. The book is written for students and researchers in statistics and machine learning, but it is also accessible to applied mathematicians and computer scientists. It includes exercises at the end of each chapter to reinforce the concepts and provide practical applications. The authors acknowledge the contributions of many collaborators and mentors, and they express gratitude for the support received from various institutions and funding bodies. Overall, the book provides a thorough and accessible introduction to statistical optimal transport, covering both theoretical foundations and practical applications. It serves as a valuable resource for those interested in understanding the role of optimal transport in statistics and its potential for advancing machine learning and data science.This text presents a comprehensive overview of statistical optimal transport, a field that combines optimal transport theory with statistical methods. The book is structured into seven main chapters and an appendix, covering foundational concepts, estimation techniques, transport maps, entropic optimal transport, Wasserstein gradient flows, and the metric geometry of the Wasserstein space. It also includes a preface, acknowledgments, and a guide on how to read the book. The book begins with an introduction to optimal transport, starting with the classical problem posed by Monge in 1781, which involves transporting sand from one location to another with minimal cost. This problem is abstracted into a problem involving probability distributions, leading to the formulation of the Kantorovich problem, which allows for more general cost functions and couplings between probability measures. The Wasserstein distance, a key concept in optimal transport, is introduced as a metric that quantifies the distance between probability distributions in a way that respects their geometry. The text then explores various aspects of optimal transport, including the estimation of Wasserstein distances, the construction of transport maps, and the use of entropic regularization. It discusses the geometric properties of the Wasserstein space, including curvature, geodesics, and gradient flows, and their applications in statistics and machine learning. The book also addresses the challenges of estimating Wasserstein distances when data are coupled, a common scenario in statistical modeling. The authors emphasize the importance of optimal transport in modern statistics and machine learning, highlighting its ability to provide a meaningful notion of distance between probability measures, which is crucial for tasks such as clustering, regression, and generative modeling. The text also discusses the computational aspects of optimal transport, including the use of the Sinkhorn algorithm and computational optimal transport, which has become a powerful tool in machine learning. The book is written for students and researchers in statistics and machine learning, but it is also accessible to applied mathematicians and computer scientists. It includes exercises at the end of each chapter to reinforce the concepts and provide practical applications. The authors acknowledge the contributions of many collaborators and mentors, and they express gratitude for the support received from various institutions and funding bodies. Overall, the book provides a thorough and accessible introduction to statistical optimal transport, covering both theoretical foundations and practical applications. It serves as a valuable resource for those interested in understanding the role of optimal transport in statistics and its potential for advancing machine learning and data science.