Abstract. Fullerenes can be modelled by cubic spherical maps maps of face-type (5,6), that is, with pentagonal and hexagonal faces only. Any such map necessaily contains exactly 12 pentagons and a hexagons, where a can be any integer number except 1. In this paper we consider an analogous problem for orientable surfaces of higher genera. Thus, we study cubic maps of face-type (6,k), where k is a "universal" value. The number of k-gonal faces depends on the genus of the surface but there is no restriction for number of hexagons. We give some general results for general cubic maps and for k=7 and 8 also for polyhedral maps. For k=7 we show that for any surface of genus g>1 and for any integer a there exists a polyhedral map of face-type (6,7) with exactly a hexagons. For k=8 we show that for any surface of genus g>1 and for any integer a there exists a polyhedral map of face-type (6,8) with exactly a hexagons; up to the case when g=2 and a≤2, when no such maps exist.