Abstract: In the 1970's, Atiyah and Singer introduced the real-valued Betti numbers associated to a normal covering space of a compact manifold. In a remark at the end of a paper, Atiyah noted that the examples that he had given all satisfied a certain integrality property and he asked for "examples where the L^2 Betti numbers are non-integral and perhaps even irrational". Despite, or because of, this formulation, the assertion that there are *no* such examples has come to be known as the 'Atiyah conjecture'.

In this talk I will survey some results on the Atiyah conjecture and its relation to the algebraic properties of group rings, and to the assembly map of geometric topology. I will not assume any prior knowledge of the technical machinery of operator algebra theory.