This book presents a comprehensive study of differential inclusions, set-valued maps, and viability theory. It is aimed at mathematicians and scientists interested in the theory and applications of differential inclusions, which generalize ordinary differential equations by allowing the velocity of a system to be a set of possible values rather than a single value. The book is divided into six chapters, each focusing on different aspects of differential inclusions and viability theory. The first chapter introduces the basic concepts of set-valued maps, including continuity, selection, and convexity. The second chapter discusses the existence of solutions to differential inclusions, with a focus on convex and nonconvex valued inclusions. The third chapter explores differential inclusions with maximal monotone maps, including the existence and uniqueness of solutions, asymptotic behavior, and gradient inclusions. The fourth chapter presents viability theory in the nonconvex case, including contingent cones, viable and monotone trajectories, and differential inclusions with memory. The fifth chapter applies viability theory to the regulation of controlled systems, focusing on convex cases and including topics such as equilibria, periodic trajectories, and price decentralization. The sixth chapter discusses Liapunov functions and their role in the existence of equilibria. The book is intended to provide a thorough understanding of differential inclusions and viability theory, with applications in various fields such as economics, social sciences, and biology. It is written for mathematicians and scientists who are interested in the theory and applications of differential inclusions and viability theory. The book is well-structured, with a clear organization of topics and a comprehensive treatment of the subject matter. It is an essential reference for anyone interested in the theory and applications of differential inclusions and viability theory.This book presents a comprehensive study of differential inclusions, set-valued maps, and viability theory. It is aimed at mathematicians and scientists interested in the theory and applications of differential inclusions, which generalize ordinary differential equations by allowing the velocity of a system to be a set of possible values rather than a single value. The book is divided into six chapters, each focusing on different aspects of differential inclusions and viability theory. The first chapter introduces the basic concepts of set-valued maps, including continuity, selection, and convexity. The second chapter discusses the existence of solutions to differential inclusions, with a focus on convex and nonconvex valued inclusions. The third chapter explores differential inclusions with maximal monotone maps, including the existence and uniqueness of solutions, asymptotic behavior, and gradient inclusions. The fourth chapter presents viability theory in the nonconvex case, including contingent cones, viable and monotone trajectories, and differential inclusions with memory. The fifth chapter applies viability theory to the regulation of controlled systems, focusing on convex cases and including topics such as equilibria, periodic trajectories, and price decentralization. The sixth chapter discusses Liapunov functions and their role in the existence of equilibria. The book is intended to provide a thorough understanding of differential inclusions and viability theory, with applications in various fields such as economics, social sciences, and biology. It is written for mathematicians and scientists who are interested in the theory and applications of differential inclusions and viability theory. The book is well-structured, with a clear organization of topics and a comprehensive treatment of the subject matter. It is an essential reference for anyone interested in the theory and applications of differential inclusions and viability theory.