Let D be a finite set of positive integers with maximum bigger than two and m̂g,n(m̃g,n) be the number on n-edged rooted maps on the orientable (nonorientable) surface of type g whose face degrees (or, dually, vertex degrees) all lie in D. Define , .
We show that m̂g(x) and m̃g(x) are algebraic functions of a certain form. Asymptotic expressions for m̂g,n and m̃g,n are also derived for some special sets D.