Regularity of Edge Ideals and Their Powers

We survey recent studies on the Castelnuovo-Mumford regularity of edge ideals of graphs and their powers. Our focus is on bounds and exact values of $\text{reg}I(G)$ and the asymptotic linear function $\text{reg}I(G{)}^q$, for $q\ge 1,$ in terms of combinatorial data of the given graph $G\mathrm{.}$