In this paper, a characterisation is given of finite
s
s
-arc transitive Cayley graphs with
s
≥
2
s\ge 2
. In particular, it is shown that, for any given integer
k
k
with
k
≥
3
k\ge 3
and
k
≠
7
k\not =7
, there exists a finite set (maybe empty) of
s
s
-transitive Cayley graphs with
s
∈
{
3
,
4
,
5
,
7
}
s\in \{3,4,5,7\}
such that all
s
s
-transitive Cayley graphs of valency
k
k
are their normal covers. This indicates that
s
s
-arc transitive Cayley graphs with
s
≥
3
s\ge 3
are very rare. However, it is proved that there exist 4-arc transitive Cayley graphs for each admissible valency (a prime power plus one). It is then shown that the existence of a flag-transitive non-Desarguesian projective plane is equivalent to the existence of a very special arc transitive normal Cayley graph of a dihedral group.