  The genus spectrum of a finite group $G$ is the set of all $g\geq 2$ such
that $G$ acts faithfully and orientation-preserving on a closed compact
orientable surface of genus $g$. This article is an overview of some results
relating the genus spectrum of $G$ to its group theoretical properties. In
particular, the arithmetical properties of genus spectra are discussed, and
explicit results are given on the 2-groups of maximal class, certain sporadic
simple groups and a some of the groups PSL$(2,q)$, where $q$ is a small prime
power. These results are partially new, and obtained through both theoretical
reasoning and application of computational techniques.
