This book deals with dilation theory of operators on Hilbert spaces and its relationship to complex geometry. Classical as well as very modern topics are covered. On the one hand, it introduces the reader to the characteristic function, a classical object used by Sz.-Nagy and Foias and still a topic of current research. On the other hand, it describes the dilation theory of the symmetrized bidisc which has been developed mostly in the present century and is a very active topic of research. It also describes an abstract theory of dilation in the setting of set theory. This was developed very recently.. A good portion of the book deals with various geometrical objects like the bidisc, the Euclidean unit ball and the symmetrized bidisc. It shows the similarities and the differences between dilation theory in these domains. While completely positive maps play a big role in dilation theory of the Euclidean unit ball, this is not so in the symmetrized bidisc for example. There, the central role is played by an operator equation. Thus, the book will introduce the reader to different techniques applicable in different domains. The first three chapters are suitable for an advanced course in operator theory in the final year of M.Sc. The whole book can be used for a first-year course for Ph.D. students. Portions of the book could also serve as material for advanced training schools in mathematics. After a course based on this book, students will be ready to work on state-of-the-art problems of research in operator theory.