This is the first existing volume that collects lectures on this important and fast developing subject in mathematics. The lectures are given by leading experts in the field and the range of topics is kept as broad as possible by including both the algebraic and the differential aspects of noncommutative geometry as well as recent applications to theoretical physics and number theory. Contents: A Walk in the Noncommutative Garden (A Connes & M Marcolli); Renormalization of Noncommutative Quantum Field Theory (H Grosse & R Wulkenhaar); Lectures on Noncommutative Geometry (M Khalkhali); Noncommutative Bundles and Instantons in Tehran (G Landi & W D van Suijlekom); Lecture Notes on Noncommutative Algebraic Geometry and Noncommutative Tori (S Mahanta); Lectures on Derived and Triangulated Categories (B Noohi); Examples of Noncommutative Manifolds: Complex Tori and Spherical Manifolds (J Plazas); D-Branes in Noncommutative Field Theory (R J Szabo).