The paper is devoted to the problem of enumerating maps on an orientable or non-orientable surface of a given genus $g$ up to all symmetries (so called unsensed maps). We obtain general formulas which reduce the problem of counting such maps to the problem of enumerating rooted quotient maps on orbifolds. In addition, we solve the problem of describing all cyclic orbifolds for a given orientable or non-orientable surface of fixed genus $g$. We also derive recurrence relations for quotient rooted maps on orbifolds that can be orientable or non-orientable surfaces with $r$ branch points, $h$ boundary components and $g$ handles or cross-caps. These results allowed us to calculate the numbers of unsensed maps on orientable or non-orientable surfaces of arbitrary genus $g$ by the number of edges.