This third edition has been critically reviewed, supplemented, and improved together with the second-named author. An additional chapter on geometric function theory and simple functions includes a proof of the Bieberbach conjecture. This second volume of function theory fully meets the expectations set by the first volume. Once again, the highly interesting historical remarks on the individual topics stand out. As a special treat, Gauss's report on Riemann's dissertation is presented. Each chapter contains detailed references. Furthermore, there are often very insightful notes on the function theory of several variables. The numerous examples and exercises provide a valuable addition to the brilliantly presented theory. The reviewer regrets that he did not have this as a student.