  By the classical genus zero Sugawara construction one obtains from admissible
representations of affine Lie algebras (Kac-Moody algebras of affine type)
representations of the Virasoro algebra. In this lecture first the classical
construction is recalled. Then, after giving a review on the global multi-point
algebras of Krichever-Novikov type for compact Riemann surfaces of arbitrary
genus, the higher genus Sugawara construction is introduced. Finally, the
lecture reports on results obtained in joint work with O.K. Sheinman. We were
able to show that also in the higher genus, multi-point situation one obtains
from representations of the global algebras of affine type representations of a
centrally extended algebra of meromorphic vector fields on Riemann surfaces.
The latter algebra is the generalization of the Virasoro algebra to higher
genus.
  Invited lecture at the XVI${}^{th}$ workshop on geometric methods in physics,
Bialowieza, Poland, June 30 -- July 6, 1997.
