A review of the first edition (1948) of this book will be found in MR0031055 (11,92e). \par The second edition appears to be about ten percent longer than the first. In his introduction, the author states that the changes consist largely of correction of errors and the reworking of certain sections for increased clarity. The author further states that ``the field of conformal mapping has not changed much since the first edition''. This ignores fifteen years of work on the numerical analysis of conformal mapping which was stimulated by electronic computers. Despite this gap, the book is now, as it was when it first appeared, a unique and useful contribution to the literature of applied mathematics. \par Table of Contents: Introduction and Simple Examples, Electric Fields, Further Examples and Consequences, General Results, Conformal Mapping in Other Areas of Physics, Conformal Mapping and Analytic Functions, Mapping by Means of Simple Functions, Some Composite Functions, Treatment of Certain Mapping Problems, Doubly Periodic Fields, Free Streams, Dictionary of Special Maps.