This very nice book is devoted to the dynamics of an iterated holomorphic mapping f from a Riemann surface to itself. The emphasis is put on the study of rational maps. As a key object, Fatou and Julia sets are defined in chapter 2. The author studies these sets on different Riemann surfaces, investigates the dynamics in the neighbourhood of fixed points and cycles and finally describes the structure of Fatou and Julia sets. In the appendices, auxiliary results and some extensions are collected. The exposition of the book is very clear. The studied problems are illustrated by many pictures and if reasonable, they are explained from different points of view. In particular, if there are independent interesting proofs for a statement, all proofs are presented. Since this is already the third edition of a book that appeared originally in 1999, many researchers in the field surely have it in their personal libraries. Those who do not will surely welcome this new edition. But the book is not just for experts in dynamics in one complex variable. Due to the clarity of presentation, it can also serve as an introduction to problems for students with a basic knowledge of complex variable theory.