This volume provides an account of the current state of the theory of combinatory spaces and discusses various applications. Here the term "combinatory space" can be regarded as a system for functional programming and bears no close connection with combinatory logic. The main chapter is divided into three chapters. Chapter 1 deals with computational structures and computability; Chapter 2 considers combinatory spaces; and Chapter 3 embraces computability in iterative combinatory spaces. A number of appendices treats a survey of examples of combinatory spaces. All sections of the chapters contain exercises together with hints for solution where appropriate. For the reading of some parts of the book a knowledge of mathematical logic and recursive function theory would be desirable. The text is mainly aimed at researchers and specialists of mathematical logic and its applications, as well as theoretical computer scientists.