  We classify the regular maps $\mathcal M$ which have automorphism groups $G$
acting faithfully and primitively on their vertices. As a permutation group $G$
must be of almost simple or affine type, with dihedral point stabilisers. We
show that all such almost simple groups, namely all but a few groups ${\rm
PSL}_2(q)$, ${\rm PGL}_2(q)$ and ${\rm Sz}(q)$, arise from regular maps, which
are always non-orientable. In the affine case, the maps $\mathcal M$ occur in
orientable and non-orientable Petrie dual pairs. We give the number of maps
associated with each group, together with their genus and extended type. Some
of this builds on earlier work of the first author on generalised Paley maps,
and on recent work of Jajcay, Li, \vSir\'a\vn and Wang on maps with
quasiprimitive automorphism groups. There are tables of data for the maps in
appendices to this paper.
