Here is a comprehensive introduction to holomorphic dynamics, that is, the dynamics induced by the iteration of various analytic maps in complex number spaces. This has been the focus of much attention in recent years, for example, with the discovery of the Mandelbrot set, and work on chaotic behavior of quadratic maps. The mathematically unified treatment emphasizes the substantial role of classical complex analysis in understanding holomorphic dynamics and offers up-to-date coverage of the modern theory. The authors cover entire functions, Kleinian groups and polynomial automorphisms of several complex variables such as complex Hénon maps, as well as the case of rational functions.