In this talk I will give a basic introduction to noncommutative geometry focusing on some of the techniques and constructions which arise in connection to questions in number theory. In the first part of the talk I will discuss Manin's real multiplication program aimed at providing a geometric framework for explicit class field theory of real quadratic fields using noncommutative tori in a way parallel to the theory of CM elliptic curves. In the second part of the talk I will discuss arithmetic quantum statistical mechanical systems whose equilibrium state structure encodes aspects of abelian class field theory, the prototypical example of these being the Bost-Connes system. In the last part of the talk I will survey some of my results and informally discuss possible relations to other approaches.