  A finite group $G$ is a DCI-group if, whenever $S$ and $S'$ are subsets of
$G$ with the Cayley graphs Cay$(G,S)$ and Cay$(G,S')$ isomorphic, there exists
an automorphism $\varphi$ of $G$ with $\varphi(S)=S'$. It is a CI-group if this
condition holds under the restricted assumption that $S=S^{-1}$. We extend
these definitions to infinite groups, and make two closely-related definitions:
an infinite group is a strongly (D)CI$_f$-group if the same condition holds
under the restricted assumption that $S$ is finite; and an infinite group is a
(D)CI$_f$-group if the same condition holds whenever $S$ is both finite and
generates $G$.
  We prove that an infinite (D)CI-group must be a torsion group that is not
locally-finite. We find infinite families of groups that are (D)CI$_f$-groups
but not strongly (D)CI$_f$-groups, and that are strongly (D)CI$_f$-groups but
not (D)CI-groups. We discuss which of these properties are inherited by
subgroups. Finally, we completely characterise the locally-finite DCI-graphs on
$\mathbb Z^n$. We suggest several open problems related to these ideas,
including the question of whether or not any infinite (D)CI-group exists.
