This monograph considers (singular) surfaces moving by mean curvature, combining tools of geometric measure theory with “viscosity solution” techniques. Employing the geometrically natural concept of “elliptic regularization”, Ilmanen establishes the existence of these surfaces. The ground-breaking work of Brakke, combined with the recently developed “level-set” approach, yields surfaces moving by mean curvature that are smooth almost everywhere. The methods developed here should form a foundation for further work in the field. This book is also noteworthy for its especially clear exposition and for an introductory chapter summarizing the key compactness theorems of geometric measure theory.