classification of Riemann surfaces for higher genus?

Question

It is well known that by uniformization theorem the simply connected Riemann surface can be classified into three equivalent classes by conformal mapping: $C, C^\infty$, and $D$. This may be seen as a generalization of the Riemann mapping theorem. And also as a general fact, the annuli in $C$ can be classified into equivalent classes of one parameter (they are all conformally equivalent to the unit disk removing a hole of radius $s$). So I'm wondering if there are any similar theorems for Riemann surfaces of the higher genus that generalizes the latter result.