Catalan numbers, named after the French-Belgian mathematician Eugène Charles Catalan (1814–1894), arise in a variety of combinatorial problems. They have many interesting properties, a rich history, and numerous arithmetic, number-theoretical, analytical, and combinatorial connections, as well as a variety of classical and modern applications. Considering the long list of open problems and questions related to the classical case, its relatives (Bell numbers, Motzkin numbers, Narayana numbers, etc.) and its generalizations, this book provides a broad perspective on the theory of this class of special numbers that will be of interest to professionals, students, and a general audience. The book begins with the history of the problem, before defining the considered numerical sets. The recurrence equation, closed formula, and generating function are then presented, followed by the simplest properties and number-theoretical properties. Later chapters discuss the relationships between Catalan numbers and other special numbers, as well as their applications and open problems. Sample Chapter(s) Preface Chapter 1: Preliminaries Contents: Preliminaries Pascal's Triangle Catalan Numbers Relatives of Catalan Numbers Catalan Numbers and Their Relatives on Integer Lattice Zoo of Numbers Mini Dictionary Exercises Readership: The target audience for this book is university teachers and students at the undergraduate and postgraduate levels interested in combinatorics, number theory, general algebra, cryptography and related fields, as well as a general audience of amateur mathematicians.