Hydrodynamic stability is of fundamental importance in fluid mechanics and is concerned with the problem of transition from laminar to turbulent flow. This book begins with a basic introduction to three major areas of the subject: thermal convection, rotating and curved flows, and parallel shear flows. There follows a comprehensive account of recent advances in the mathematical theory for parallel shear flows. A number of applications of the linear theory are discussed, including the effects of stratification and unsteadiness. The book concludes with a chapter describing some of the fundamental ideas involved in current work on nonlinear hydrodynamic stability. The emphasis throughout is on the ideas involved, the physical mechanisms, the methods used, and the results obtained, and, wherever possible, the theory is related to both experimental and numerical results. A distinctive feature of the book is the large number of problems it contains. These problems, for which hints and references are given, not only provide exercises for students but also provide many additional results in a concise form.