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