This book is an expanded version of talks presented at a conference held at Santa Barbara in 2003 to celebrate the 75th birthday of Professor H. S. Shapiro. Quadrature domains are related to Gaussian type quadratures for classes of integrable analytic or harmonic functions. The origin goes back to specific problems of classical potential theory. However, the research developed intensively since the seventies has an interdisciplinary character: univalent functions, approximation theory, fluid mechanics, variational problems for partial differential equations, free boundaries, etc. The introductory article, ‘What is a quadrature domain’, written by B. Gustafsson and H. S. Shapiro, gives an informative introduction to the subject. The remaining twelve papers cannot be described in detail here. One deals with fluid dynamics, one with the Brownian motion aspects of quadrature domains, four papers are associated to potential theory and six papers are linked to complex analysis. A selected bibliography by H. S. Shapiro and an open problems section are included. The book gives a nice overview of recent achievements in the field of quadrature domains.

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