This book develops a variety of aspects of analysis and geometry on foliated spaces which should be useful in many situations. These strands are then brought together to provide a context and to expose Connes index theorem for foliated spaces, a theorem which asserts the equality of the analytic and the topological index (two real numbers) which are associated to a tangentially elliptic operator. An additional purpose of this exposition is preparing the way towards the more general index theorem of Connes and Skandalis. This index theorem describes the abstract index class in KO (CR*(G(M))),...