Written by Luis Caffarelli and Sandro Salsa, this book offers an excellent exposition on free boundary problems. Free or moving boundary problems appear in many areas of analyis, geometry, and applied mathematics. A typical example is the evolving interphase between a solid and liquid phase: if we know the initial configuration well enough, we should be able to reconstruct its evolution, in particular, the evolution of the interphase. In this book the authors present a series ofideas, methods, and techniques for treating the most basic issues of such a problem. In particular, they describe the very fundamental tools of geometry and real analysis that make this possible: properties of harmonic and caloric measure in Lipschitz domains, a relation between parallel surfaces andelliptic equations, monotonicity formulas and rigidity, etc. The tools and ideas presented here will serve as a basis for the study of more complex phenomena and problems. This book is useful for supplementary reading or will be a fine independent study text. It is suitable for graduate students and researchers interested in partial differential equations. Also available from the AMS by Luis Caffarelli is Fully Nonlinear Elliptic Equations