This book is an attempt to cover some of the salient features of classical, one variable complex function theory. The approach is analytic, as opposed to geometric, but the methods of all three of the principal schools (those of Cauchy, Riemann and Weierstrass) are developed and exploited. The book goes deeply into several topics (e.g. convergence theory and plane topology), more than is customary in introductory texts, and extensive chapter notes give the sources of the results, trace lines of subsequent development, make connections with other topics and offer suggestions for further reading. These are keyed to a bibliography of over 1300 books and papers, for each of which volume and page numbers of a review in one of the major reviewing journals is cited. These notes and bibliography should be of considerable value to the expert as well as to the novice. For the latter there are many references to such thoroughly accessible journals as the American Mathematical Monthly and L'Enseignement Math?matique. Moreover, the actual prerequisites for reading the book are quite modest; for example, the exposition assumes no fore knowledge of manifold theory, and continuity of the Riemann map on the boundary is treated without measure theory. "This is, I believe, the first modern comprehensive treatise on its subject. The author appears to have read everything, he proves everything and he has brought to light many interesting but generally forgotten results and methods. The book should be on the desk of everyone who might ever want to see a proof of anything from the basic theory. ..." (SIAM Review) / " ... An attractive ingenious and many time humorous form increases the accessibility of the book. ..." (Zentralblatt f?r Mathematik) / "Professor Burckel is to be congratulated on writing such an excellent textbook. ... this is certainly a book to give to a good student and he would profit immensely from it. ..." (Bulletin London Mathematical Society)