Every finite group acts as a group of automorphisms of some compact bordered Klein surface of algebraic genus g ≥ 2. The same group G may act on different genera and so it is natural to look for the minimum genus on which G acts. This is the minimum genus problem for the group G. On the other hand, for a fixed integer g ≥ 2, there are finitely many abstract groups acting as a group of automorphisms of some compact bordered Klein surface of algebraic genus g. The condition g ≥ 2 assures that all such groups are finite. So it makes sense to look for the largest order of groups G acting on some surface of genus g when g is fixed and G runs over a prescribed family F of groups. This is the maximum order problem for the family F. There is a significant amount of research dealing with these two problems (or with some of their variations), and the corresponding results are scattered in the literature. The purpose of this survey is to gather some of these results, paying special attention to important families of finite groups.
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