  We discuss recent progress on the problem of classifying point-primitive
generalised polygons. In the case of generalised hexagons and generalised
octagons, this has reduced the problem to primitive actions of almost simple
groups of Lie type. To illustrate how the natural geometry of these groups may
be used in this study, we show that if $\mathcal{S}$ is a finite thick
generalised hexagon or octagon with $G \leqslant{\rm Aut}(\mathcal{S})$ acting
point-primitively and the socle of $G$ isomorphic to ${\rm PSL}_n(q)$ where $n
\geqslant 2$, then the stabiliser of a point acts irreducibly on the natural
module. We describe a strategy to prove that such a generalised hexagon or
octagon $\mathcal{S}$ does not exist.
