This book is concerned with the theory of gravity–capillary free-surface flows. Free-surface flows are flows bounded by surfaces that have to be found as part of the solution. A canonical example is that of waves propagating on a water surface, the latter in this case being the free surface.

Many other examples of free-surface flows are considered in the book (cavitating flows, free-surface flows generated by moving disturbances, rising bubbles etc.). I hope to convince the reader of the beauty of such problems and to elucidate some mathematical challenges faced when solving them. Both analytical and numerical methods are presented. Owing to space limitations, some topics could not be covered. These include interfacial flows and the effects of viscosity, compressibility and surfactants. Some further developments of the theories described in the book can be found in the list of references.

Many results presented in the book have grown out of my research over the last 35 years and, of course, out of the research of the whole fluid mechanics community. References to the original papers are given. For this book, I have repeated the older numerical calculations with larger numbers of grid points than was possible at the time. I am pleased to report that the new results are in agreement with the earlier ones!