The first edition of this influential book, published in 1970, opened up a completely new field of invariant metrics and hyperbolic manifolds. This book continues to serve as the best introduction to hyperbolic complex analysis and geometry and is easily accessible to students since very little is assumed. Contents: The Schwarz Lemma and Its Generalizations; Volume Elements and the Schwarz Lemma; Distance and the Schwarz Lemma; and more.