0 Introduction to M ( atrix ) theory and noncommutative geometry

We give a mostly self-contained review of some aspects of M(atrix) theory and noncommutative geometry. The topics include introduction to BFSS and IKKT matrix models, compactifications on noncommutative tori, a review of basic notions of noncommutative geometry with a detailed discussion of noncommutative tori, Morita equivalence and SO(d, d|Z)-duality, an elementary discussion of instantons and noncommutative orbifolds. The review is primarily intended for physicists who would like to learn some basic techniques of noncommutative geometry and how they can be applied in string theory and for mathematicians who would like to learn about some new problems arising in theoretical physics.