We study the cohomology of various Lie algebras and differential graded
Lie algebras of vector fields on Riemann surfaces. Topics include the
Kodaira-Spencer differential graded Lie algebra and its relation to
deformation theory, its sub Lie algebra of smooth vector fields of
anti-holomorphic type and a 2 dimensional analogue of the Virasoro
algebra, and finally Krichever-Novikov algebras being Lie algebras
of
meromorphic vector fields with poles in a fixed set of points.
In conclusion we state some open problems.