The classical Local Langlands correspondence sets up a bijection between (isomorphism classes of) n-dimensional representations of the Weil-Deligne group of a given local field F, and (isomorphism classes of) irreducible smooth representations of GL(n,F).

This correspondence has the property that the representations on the Galois side under consideration are trivial on an open subgroup of the wild inertia group.

In particular, this correspondence neglects almost all Galois representations which are p-adically continuous (p being the residue characteristic of F).

In recent years a picture emerged how one may possibly extend this correspondence to include certain p-adic Galois representations. The objects on the automorphic side are continuous representations of GL(n,F) on Banach spaces.

In this talk we will give an overview of the basic ideas behind this p-adic Langlands correspondence, discuss some results, and explain a construction of Banach space completions of supercuspidal representations of GL(2,F) (the latter being joint work in progress with C. Breuil).