Cyclic $n$-gonal Surfaces

A cyclic $n$-gonal surface is a compact Riemann surface $X$ of genus $g\geq 2$admittingacyclicgroupofconformalautomorphisms$C$oforder$n$ such that the quotient space $X\mathrm{/}C$ has genus 0. In this paper, we provide an overview of ongoing research into automorphism groups of cyclic $n$-gonal surfaces. Much of the paper is expository or will appear in forthcoming papers, so proofs are usually omitted. Numerous explicit examples are presented illustrating the computational methods currently being used to study these surfaces.