In their seminal 1995 paper, Bost and Connes constructed, for certain inclusions of discrete groups H \subset G, a C*-algebra which is a completion of the Hecke algebra of bi-H-invariant functions on G. This C*-algebra carries with it a natural time evolution measuring the lack of normality of H in G. The resulting C*-dynamical system, applied to a certain inclusion of ax+b groups, is called the Bost-Connes system. In this talk, I will give an introduction to the Bost-Connes system and its fascinating thermodynamic properties, which are intimately related to number theory.