What is a Quadrature Domain? -- Recent Progress and Open Problems in the Bergman Space -- The Bergman Kernel and Quadrature Domains in the Plane -- The Cauchy Transform -- Quadrature Domains and Fluid Dynamics -- On Uniformly Discrete Sequences in the Disk -- Algebraic Aspects of the Dirichlet Problem -- Linear Analysis of Quadrature Domains. IV -- Restriction, Localization and Microlocalization -- Quadrature Domains and Brownian Motion (A Heuristic Approach) -- Weighted Composition Operators Associated with Conformal Mappings -- Quadrature Identities and Deformation of Quadrature Domains -- Subharmonicity of Higher Dimensional Exponential Transforms.Quadrature domains were singled out about 30 years ago by D. Aharonov and H.S. Shapiro in connection with an extremal problem in function theory. Since then, a series of coincidental discoveries put this class of planar domains at the center of crossroads of several quite independent mathematical theories, e.g., potential theory, Riemann surfaces, inverse problems, holomorphic partial differential equations, fluid mechanics, operator theory. The volume is devoted to recent advances in the theory of quadrature domains, illustrating well the multi-facet aspects of their nature. The book contains a large collection of open problems pertaining to the general theme of quadrature domains.