The warm reception with which the first edition of this book has been received has been a source of both pride and pleasure. It is a special privilege to have created a “for the record” version of Ahlfors's seminal ideas in the subject. And the geometric viewpoint continues to develop.

In the intervening decade, this author has learned a great deal more about geometric analysis, and his view of the subject has developed and broadened. It seems appropriate, therefore, to bring some new life to these pages, and to set forth a fresh enunciation of the role of curvature in basic complex function theory.

In this new edition, we explain how, in a natural and elementary manner, the hyperbolic disc is a model for the non-Euclidean geometry of Bolyai and Lobachevsky. Later on, we explain the Bergman kernel and provide an introduction to the Bergman metric.

I have many friends and colleagues to thank for their incisive remarks and suggestions about the first edition of this book. I hope that I do them justice in my efforts to implement a second edition. As always, the Mathematical Association of America has been an exemplary publisher and has provided all possible support in the publication process. I offer my humble thanks.