  Affine Kac-Moody algebras give rise to interesting systems of differential
equations, so-called Knizhnik-Zamolodchikov equations. The monodromy properties
of their solutions can be encoded in the structure of a modular tensor category
on (a subcategory of) the representation category of the affine Lie algebra. We
discuss the relation between these solutions and physical correlation functions
in two-dimensional conformal field theory. In particular we report on a proof
for the existence of the latter on world sheets of arbitrary topology.
