Amenable Operator Algebras

Operator algebras come in two distinct flavours --- C*-algebras and von Neumann algebras --- which provide frameworks for the study of non-commutative topology and measure theory. In the 1970's Connes imported ideas originating in amenability for groups into the setting of von Neumann algebras, leading to a complete classification of amenable von Neumann algebras.  The last decade has seen dramatic progress on the structure and classification of simple amenable C*-algebras, with suprising connections being made to Coness work.  I'll survey some of this progress; no prior knowledge of operator algebras will be assumed.