Our purpose and main concern in writing this book is to illuminate classical concepts from the noncommutative viewpoint, to make the language and techniques of noncommutative geometry accessible and familiar to practi tioners of classical mathematics, and to benefit physicists interested in the uses of noncommutative spaces. Same may say that ours is a very "com mutative" way to deal with noncommutative matters; this charge we readily admit. Noncommutative geometry amounts to a program of unification of math ematics under the aegis of the quantum apparatus, i.e., the theory of ope rators and of C -algebras. Largely the creation of a single person, Alain Connes, noncommutative geometry is just coming of age as the new century opens. The bible of the subject is, and will remain, Connes' Noncommuta tive Geometry (1994), itself the "3.8-fold expansion" of the French Geome trie non commutative ( 1990). Theseare extraordinary books, a "tapestry" of physics and mathematics, in the words of Vaughan jones, and the work of a "poet of modern science," according to Daniel Kastler, replete with subtle knowledge and insights apt to inspire several generations.

"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