Abstract: Group coloring is a generalization of vertex coloring of graphs. For a digraph $G$ and a group $\Gamma$, we consider an assignment $f : E(G) \to \Gamma$ of group elements to arcs and then attempt to color the vertices with a function $c : V(G) \to \Gamma$ in such a way that the colors of two adjacent vertices do not differ by the value assigned to the arc between them (i.e. $c(u)c(v)^{-1} \neq f(uv)$ for $uv \in E(G)$). Unlike in classical vertex coloring problems, it is meaningful to consider group coloring (and some other even more general coloring notions) of multigraphs. In this talk I will present some recent results and open conjectures involving the coloring of multigraphs, particularly some local structures that have been useful in proving these results and working towards resolving the conjectures.