The work of von Neumann established that the analytic structure of a group is encoded by the structure of a corresponding algebra of operators. I will give an introduction to recent work along these lines which makes a surprising connection to the topological ergodic theory of Furstenberg, and I will discuss how these ideas were used to settle several longstanding open problems in analytic group theory. It is natural to seek to extend these ideas to the more general setting of a quantum group, which provides the setting for non- commutative Fourier analysis. I will briefly mention ongoing work in this direction, as well as some applications to quantum information theory. This is joint work with Emmanuel Breuillard, Matthew Kennedy and Narutaka Ozawa.