  For each finite classical group $G$, we classify the subgroups of $G$ which
act transitively on a $G$-invariant set of subspaces of the natural module,
where the subspaces are either totally isotropic or nondegenerate. Our proof
uses the classification of the maximal factorisations of almost simple groups.
As a first application of these results we classify all point-transitive
subgroups of automorphisms of finite thick generalised quadrangles.
