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This monograph provides a comprehensive and accessible introduction to the optimization of elliptic systems. The book addresses some of the pressing unsolved questions in the field, concentrating on two main directions: the optimal control of linear and nonlinear elliptic equations, and problems involving unknown and/or variable domains. Throughout, the authors elucidate connections between seemingly different types of problems. One basic feature is to relax the needed regularity assumptions as much as possible to include larger classes of possible applications. The six chapters give a gradual and accessible presentation of the material, and a special effort is made to present numerous examples. In addition to its appeal to students and researchers, much of this material will prove useful for scientists from other fields where the optimization of elliptic systems occurs, such as physics, mechanics, and engineering.