Schlichenmaier Martin Algebras of Vector Fields on Riemann Surfaces (Generalized Krichever - Novikov - Algebras) (38K, amstex 1.1) ABSTRACT. Generalizations of the Krichever - Novikov algebras (resp. of the Virasoro algebra) in the case of many punctures are introduced. This is of importance in Conformal Field Theory. Their representations and central extensions are studied. (This is a wirte-up of a talk)