Free-surface problems occur in many aspects of science and everyday life. They can be defined as problems whose mathematical formulation involves surfaces that have to be found as part of the solution. Such surfaces are called free surfaces. Examples of free-surface problems are waves on a beach, bubbles rising in a glass of champagne, melting ice, flows pouring from a container and sails blowing in the wind. In these examples the free surface is the surface of the sea, the interface between the gas and the champagne, the surface of the ice, the boundary of the pouring flow and the surface of the sail.

In this book we concentrate on applications arising in fluid mechanics. We hope to convince the reader of the beauty of such problems and to present the challenges faced when one attempts to describe these flows mathematically. Many of these challenges are resolved in the book but others are still open questions. We will always attempt to present fully nonlinear solutions without restricting assumptions on the smallness of some parameters. Our techniques are often numerical. However, it is the belief of the author that a deep understanding of the structure of the solutions cannot be gained by brute-force numerical approaches. It is crucial to combine numerical methods with analytical techniques, especially when singularities are present.