In the study of short-wave diffraction problems, asymptotic methods - the ray method, the parabolic equation method, and its further development as the "etalon" (model) problem method - play an important role. These are the methods to be treated in this book. The applications of asymptotic methods in the theory of wave phenomena are still far from being exhausted, and we hope that the techniques set forth here will help in solving a number of problems of interest in acoustics, geophysics, the physics of electromagnetic waves, and perhaps in quantum mechanics. In addition, the book may be of use to the mathematician interested in contemporary problems of mathematical physics. Each chapter has been annotated. These notes give a brief history of the problem and cite references dealing with the content of that particular chapter. The main text mentions only those publications that explain a given argument or a specific calculation. In an effort to save work for the reader who is interested in only some of the problems considered in this book, we have included a flow chart indicating the interdependence of chapters and sections. The authors consider it their pleasant duty to thank M.M. Popov, who, at the authors' request, wrote Sects. 10.1-7, and also LA. Molotkov, who was involved in all stages of preparing this book. LA. Molotkov wrote Sect. 7.5 and most of Chap. 11, and was our coauthor in writing Chap. 6. We are indebted to participants at the seminar of the Leningrad Section of the V.A. Steklov Institute of Mathematics and Leningrad State University on the mathematical theory of diffraction, but especially to V.F. Lazutkin, for their constructive criticism. We owe a great deal to the editors of the book. Their careful work, at times going far beyond the limits of direct editorial duties, enabled us to eliminate a number of shortcomings in the manuscript