Abstract: Let S be a finite thick generalised hexagon or octagon. If S admits a group G of automorphisms acting distance transitively, then G is known, and S is a known `classical’ generalised polygon. We consider weakening the assumption of a distance-transitive group action to a group acting primitively on the points of S . A result of Bamberg, Glasby, Popiel, Praeger and Schneider tells us that in this case the socle of G is a finite simple group of Lie type. I shall present some progress on how much more can be said. This is joint work in the case of the classical groups with Glasby and Praeger and with Popiel in the case of exceptional groups of Lie type.