  In this paper, we study arc-transitive Cayley graphs on non-abelian simple
groups with soluble stabilizers and valency seven. Let $\Ga$ be such a Cayley
graph on a non-abelian simple group $T$. It is proved that either $\Ga$ is a
normal Cayley graph or $\Ga$ is $S$-arc-transitive, with
$(S,T)=(\A_n,\A_{n-1})$ and $n=7,21,63$ or $84$; and, for each of these four
values of $n$, there really exists arc-transitive $7$-valent non-normal Cayley
graphs on $\A_{n-1}$ and specific examples are constructed.
