polynomial ring in $n$ variables over $\mathbb{K}$. Suppose that $M$ is a graded $S$-module with minimal free resolution

$$0 \longrightarrow \cdots \longrightarrow \bigoplus_{j}S(-j)^{\beta_{1,j}(M)} \longrightarrow \bigoplus_{j}S(-j)^{\beta_{0,j}(M)} \longrightarrow M \longrightarrow 0.$$

The Castelnuovo--Mumford regularity (or simply, regularity) of $M$, denote by ${\rm reg}(M)$, is defined as follows:

$${\rm reg}(M)={\rm max}\{j-i|\ \beta_{i,j}(M)\neq0\}.$$

We survey a number of recent studies of the Castelnuovo-Mumford regularity of the ideals related to a graph and their (symbolic) powers. Our focus is on the bounds and exact values for the regularity in terms of combinatorial data from associated graphs. This research program has produced many exciting results and, at the same time, opened many further interesting questions and conjectures.