|This book makes available a comprehensive, detailed and carefully organized treatment of the foundations for the theory of iteration of rational functions of a complex variable. The material extends from the orginial memoirs of Fatou and Julia to the recent and important results and methods of Sullivan and Shishikura. Many of the details of the proofs have not occurred in print before. The theory of dynamical systems and chaos has recently undergone a rapid growth in popularity, in part due to the spectacular computer graphics of Julia sets, fractals, and the Mandelbrot set. This text focuses on the specialized area of complex analytic dynamics, a subject which dates back to 1916 and is currently a very active area in mathematics.