  In this paper we investigate orders, longest cycles and the number of cycles
of automorphisms of finite vertex-transitive graphs. In particular, we show
that the order of every automorphism of a connected vertex-transitive graph
with $n$ vertices and of valence $d$, $d\le 4$, is at most $c_d n$ where
$c_3=1$ and $c_4 = 9$. Whether such a constant $c_d$ exists for valencies
larger than $4$ remains an unanswered question. Further, we prove that every
automorphism $g$ of a finite connected $3$-valent vertex-transitive graph
$\Gamma$, $\Gamma \not\cong K_{3,3}$, has a regular orbit, that is, an orbit of
$\langle g \rangle$ of length equal to the order of $g$. Moreover, we prove
that in this case either $\Gamma$ belongs to a well understood family of
exceptional graphs or at least $5/12$ of the vertices of $\Gamma$ belong to a
regular orbit of $g$. Finally, we give an upper bound on the number of orbits
of a cyclic group of automorphisms $C$ of a connected $3$-valent
vertex-transitive graph $\Gamma$ in terms of the number of vertices of $\Gamma$
and the length of a longest orbit of $C$.
