Developed by Alain Connes, noncommutative geometry is the set of tools and methods that makes possible the classification and analysis of a broad range of objects beyond the reach of classical methods. This English version of the author's path-breaking French book on the subject gives the definitive treatment of his revolutionary approach to measure theory, geometry, and mathematical physics. Profusely illustrated and invitingly written, this book is ideal for anyone who wants to know what noncommutative geometry is, what it can do, or how it can be used in various areas of mathematics, quantization, and elementary particles and fields.