  In this paper we discuss a method for bounding the size of the stabiliser of
a vertex in a $G$-vertex-transitive graph $\Gamma$. In the main result the
group $G$ is quasiprimitive or biquasiprimitive on the vertices of $\Gamma$,
and we obtain a genuine reduction to the case where $G$ is a nonabelian simple
group.
  Using normal quotient techniques developed by the first author, the main
theorem applies to general $G$-vertex-transitive graphs which are $G$-locally
primitive (respectively, $G$-locally quasiprimitive), that is, the stabiliser
$G_\alpha$ of a vertex $\alpha$ acts primitively (respectively
quasiprimitively) on the set of vertices adjacent to $\alpha$. We discuss how
our results may be used to investigate conjectures by Richard Weiss (in 1978)
and the first author (in 1998) that the order of $G_\alpha$ is bounded above by
some function depending only on the valency of $\Gamma$, when $\Gamma$ is
$G$-locally primitive or $G$-locally quasiprimitive, respectively.
