Free or moving boundary problems appear in many areas of analysis, 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 of ideas, 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 measures in Lipschitz domains, a relation between parallel surfaces and elliptic 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.