Finite $s$-arc transitive Cayley graphs and flag-transitive projective planes

In this paper, a characterisation is given of finite $s$-arc transitive Cayley graphs with $s\ge2$. In particular, it is shown that, for any given integer $k$ with $k\ge3$ and $k\not=7$, there exists a finite set (maybe empty) of $s$-transitive Cayley graphs with $s\in\{3,4,5,7\}$ such that all $s$-transitive Cayley graphs of valency $k$ are their normal covers. This indicates that $s$-arc transitive Cayley graphs with $s\ge3$ 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.