  This is a survey of our construction of current algebras, associated with
complex curves and rational differentials. We also study in detail two classes
of examples. The first is the case of a rational curve with differentials $z^n
dz$; these algebras are ``building blocks'' for the quantum current algebras
introduced in our earlier work. The second is the case of a genus $>1$ curve
$X$, endowed with a regular differential having only double zeroes.
