The main theme of this book is the study of geometric properties of general sets and measures in euc lidean space. Examples to which this theory applies include fractal-type objects such as strange attractors for dynamical systems, and those fractals used as models in the sciences.The author provides a firm and unified foundation for the subject and develops all the main tools used in its study, such as covering theorems, Hausdorff measures and their relations to Riesz capacities and Fourier transforms. Thus the book is essentially self-contained for graduate students in mathematics; it is primarily targetted at them and researchers.The last third of the book is devoted to the Besicovitch-Federer theory of rectifiable sets, which form in a sense the largest class of subsets of euclidean space possessing many of the properties of smooth surfaces. These sets have wide application; for example they are central in the higher-dimensional calculus of variations. Their relations to complex analysis and singular integrals are also studied.