Random sets play an important role in many applications of mathematics and have been studied intensively since the 1970s, when the pioneering book of Georges Matheron was published. This book is an important contribution to the mathematical theory and will surely serve as a valuable textbook for students as well as researchers. It presents a self-contained survey of all the significant results, including the proofs about random sets and capacity functionals, convergence of random sets, expectations, limit laws for Minkowski additions and for unions, and stochastic processes of random closed sets. The theory is illustrated by examples of applications, e.g. in stochastic geometry, stochastic optimisation, belief functions and finance mathematics. A number of open problems are presented and each chapter concludes with a list of bibliographical notes.