Given a group $G$ and $S \subset G$, the Cayley (di)graph Cay$(G,S)$ is the graph whose vertex set is $G$ with an arc from $g$ to $h$ if and only if $hg^{-1}\in S$. Although the graph isomorphism problem in general is only known to be quasipolynomial, for some families of Cayley (di)graphs it reduces to understanding the automorphisms of the groups that are involved. The question of when this happens is known as the Cayley Isomorphism (CI) problem.
I will provide an overview of the current state of the CI problem for both finite and infinite groups.