  We introduce C-Algebras of compact Riemann surfaces $\Sigma$ as
non-commutative analogues of the Poisson algebra of smooth functions on
$\Sigma$. Representations of these algebras give rise to sequences of
matrix-algebras for which matrix-commutators converge to Poisson-brackets as
$N\to\infty$. For a particular class of surfaces, nicely interpolating between
spheres and tori, we completely characterize (even for the intermediate
singular surface) all finite dimensional representations of the corresponding
C-algebras.
