An orientably regular map is a 2-cell decomposition of a closed orientable surface with the largest possible number of orientation-preserving symmetries; equivalently, its orientation-preserving automorphism group acts transitively on the set of oriented edges (or arcs, or darts). The finite group \(G\) of orientation-preserving automorphisms of such a regular map is a finite quotient of the free product \(\mathbb{Z}\ast\mathbb{Z}_ 2\) or, equivalently, of a triangle group of type \((2,p,q)\) (corresponding to a regular map with \(p\)-gonal faces, with \(q\) faces meeting at each vertex). The main subject addressed in the present paper is the classification of the regular maps with nilpotent automorphism group \(G\) (“nilpotent regular maps”); by known results, the main point here is the problem of classifying nilpotent maps with simple underlying graph (without multiple edges). “Our main objective in this paper is to show that for any positive integer \(c\), there are only finitely many regular maps with simple underlying graph, such that the orientation-preserving automorphism group of the map is nilpotent of class \(c\). In fact, we prove that the number of vertices of any such graph is bounded by a function of \(c\). Also, we give an exact formula for the maximum number of vertices of a simple nilpotent map of class \(c\), and show that this maximum is achieved by exactly one nilpotent simple map of which every other simple nilpotent map of the given class \(c\) is a quotient.” Finally, a list of all nilpotent maps of classes 2 to 4 is given together with the order and a presentation of \(G\) and the type \((p,q)\) of a map (obtained by computer).