Is there a "p-adic analytic Langlands correspondence" which gives an automorphic interpretation to the families of overconvergent modular forms and p-adic Galois representations that have been constructed by Hida and Coleman? In this talk I will survey some recent developments suggesting that such a correspondence exists. In particular I will give an overview of the theory of p-adic analytic representation theory I have been developing with Peter Schneider and discuss briefly how it has been related to arithmetic by Breuil and Emerton.