"In collecting the material in a coherent form, the authors have performed a valuable service and this book will no doubt open noncommutative geometry to a wider audience... [A] large number of metaphorical and mathematical remarks embedded in the text...highlight the plethora of connections between the material of the book and other areas of mathematics." -Mathematical Reviews "The material is collected from an enormous number of sources... All of it is carefully worked out and serves as a highly recommendable introduction to the vital field of noncommutative geometry." -Zentralblatt Math "The style of the book is lively, exempt of the stereotypes one encounters rather often in the mathematical literature. The authors know how to tell stories and enjoy doing it. The friendly tone could be of some help for an hypothetical unexperienced reader confronted with almost 700 pages of rather difficult and delicate mathematics." -Journal of Operator Theory "The present book is a systematic course in noncommutative differential geometry and operator theory, with applications to guantum physics. Its topics cover C*-algebras, vector bundles and C*-modules, K-theory, Fredholm operators, Clifford algebras, Spin representations, noncommutative integration and differential calculus, spectral triples and Connes spin manifold theorem. As applications, noncommutative tori, quantum theory and Kreimer-Connes-Moscovici algebras are discussed. The book will be helpful to all mathematicians and mathematical physicits who wish to learn about noncommutative geometry and tis ramifications." ---Matheamtica Bohemica