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Free surface problems occur in many aspects of science and of everyday life such as the waves on a beach, bubbles rising in a glass of champagne, melting ice, pouring flows from a container and sails billowing in the wind. Consequently, the effect of surface tension on gravity-capillary flows continues to be a fertile field of research in applied mathematics and engineering. Concentrating on applications arising from fluid dynamics, Vanden-Broeck draws upon his years of experience in the field to address the many challenges involved in attempting to describe such flows mathematically. Whilst careful numerical techniques are implemented to solve the basic equations, an emphasis is placed upon the reader developing a deep understanding of the structure of the resulting solutions. The author also reviews relevant concepts in fluid mechanics to help readers from other scientific fields who are interested in free boundary problems.