Krichever-Novikov type algebras and Wess-Zumino-Witten models

Krichever--Novikov type algebras are generalizations of the Witt,
Virasoro, affine Lie algebras, and their relatives to Riemann
surfaces of arbitrary genus and/or the multi-point situation.
They play a very important role in the context of
quantization of Conformal Field Theory. In this review
we give the most important results about their structure,
almost-grading and central extensions.
Furthermore, we explain how they are used in the context
of Wess--Zumino--Novikov--Witten models,
respectively Knizhnik-Zamolodchikov connections.
There they play a role
as gauge algebras,
as tangent directions to the moduli spaces,
and as Sugawara operators.