This carefully written textbook is an introduction to the beautiful concepts and results of complex analysis. It is intended for international bachelor and master programmes in Germany and throughout Europe; in the Anglo-American system of university education the content corresponds to a beginning graduate course. The book presents the fundamental results and methods of complex analysis and applies them to a study of elementary and non-elementary functions (elliptic functions, Gamma- and Zeta function including a proof of the prime number theorem ...) and - a new feature in this context! - to exhibiting basic facts in the theory of several complex variables. Part of the book is a translation of the authors' German text Einfuhrung in die komplexe Analysis; some material was added from the by now almost classical text Funktionentheorie written by the authors, and a few paragraphs were newly written for special use in a master's programme.