Locally semialgebraic spaces serve as an appropriateframework for studying the topological properties ofvarieties and semialgebraic sets over a real closed field. This book contributes to the fundamental theory ofsemialgebraic topology and falls into two main parts. The first dealswith sheaves and their cohomology on spaceswhich locally look like a constructible subset of a realspectrum. Topics like families of support, homotopy, acyclicsheaves, base-change theorems and cohomological dimensionare considered. In the second part a homology theory for locally completelocally semialgebraic spaces over a real closed field isdeveloped, the semialgebraic analogue of classical Bore-Moore-homology. Topics include fundamental classes ofmanifolds and varieties, Poincare duality, extensions of thebase field and a comparison with the classical theory. Applying semialgebraic Borel-Moore-homology, a semialgebraic("topological") approach to intersection theory on varietiesover an algebraically closed field of characteristic zero isgiven. The book is addressed to researchers and advancedstudents in real algebraic geometry and related areas.