The classical Langlands correspondence relates representations of a reductive algebraic group over a local non-archimedian field $F$ and representations of the Galois group of $F$. If we replace $F$ by the field $C((t))$ of complex Laurent power series, then the corresponding group becomes the (formal) loop group. It is natural to ask: is there an analogue of the Langlands correspondence in this case? It turns out that the answer is affirmative, and there is an interesting theory which may be viewed as both "geometrization" and "categorification" of the classical theory. I will explain the general set-up for this new theory and give some examples using representations of affine Kac-Moody algebras.