The importance of the potential surface model has led naturally to a large number of studies on the subject, where the emphasis has usually been placed on lower dimensional problems, such as the reaction dynamics of diatomic to four-atom systems, or conformational problems restricted to few internal rotations. The purposes and methods of this book are, however, somewhat different from those of most studies on potential surface problems. The emphasis here is placed on those fundamental properties of potential energy hypersurfaces that are general for higher dimensions, that is, for larger molecules. The study of these properties requires some of the tools of global analysis that are not among the routine mathematical techniques of quantum chemists: topology, homotopy, and homology. This book provides the reader with an introduction to the fundamentals and to some of the more recent developments in the theory of potential energy hypersurfaces. The text is fairly self-contained. It requires no previous mathematical knowledge from the reader beyond that needed in an undergraduate quantum chemistry course.