Abstract. Mathematical models of fullerenes are cubic spherical and polyhedral maps of face type (5,6), that is, with pentagonal and hexagonal faces only. Any such map necessarily contains 12 pentagons and it is known that for any integer a≥0 except a=1 there exists a fullerene map with precisely a hexagons. Here we consider hyperbolic analogues of fullerenes modelled by cubic polyhedral maps of face-type (6,k), where 9≤k≤10, on orientable surface of genus at least two. The number of k-gons depends on the genus but the number of hexagons is again independent of the surface. For every triple k,g,a, where 9≤k≤10, g≥2 and a≥0, we determine if there exists a cubic polyhedral map of face-type (6,k) with exactly a hexagons on an orientable surface of genus g. The only unsolved cases are k=10, g=5 and a≤3 when we are not able to say if a hyperbolic fullerene with these parameters exist.