Multivalent and in particular univalent functions play an important role in complex analysis. Great interest was aroused when de Branges in 1985 settled the long-standing Bieberbach conjecture for the coefficients of univalent functions. The second edition of Professor Hayman's celebrated book is the first to include a full and self-contained proof of this result, with a new chapter devoted to it. Another new chapter deals with coefficient differences of mean p-valent functions. The book has been updated in several other ways, with recent theorems of Baernstein and Pommerenke on univalent functions of restricted growth and Eke's regularity theorems for the behaviour of the modulus and coefficients of mean p-valent functions. Some of the original proofs have been simplified. Each chapter contains examples and exercises of varying degrees of difficulty designed both to test understanding and to illustrate the material. Consequently the book will be useful for graduate students and essential for specialists in complex function theory.