Abstract: Operator algebras arise as suitably closed subalgebras of the bounded operators on a Hilbert space. They come in two distinct types: von Neumann algebras which have the flavour of measure theory, and C*-algebras which have the flavour of topology. In the 1970?s Alain Connes obtained a deep structural theorem for amenable von Neumann algebras, leading to a complete classification of these objects. For the last 25 years the Elliott classification programme has been seeking a corresponding result for simple amenable C*-algebras, and now, though the efforts of numerous researchers worldwide, we have a definitive classification theorem. In this talk, I?ll explain what this theorem says, and the analogies it makes to Connes work, using examples from groups and dynamics as motivation.