For a finite set \(D\) of positive integers with maximum exceeding 2 the author considers rooted maps with a root face and other “distinguished” faces “marked” by the distinct positive integers of a set \(I\); \(z_ I\) is a set of indeterminates indexed by \(I\); all faces except the root and distinguished faces have degree in \(D\). The generating functions for numbers of such rooted maps on orientable and nonorientable surfaces of type \(g=1-{12 \over \chi}\), where \(x\), \(y\), \(z_ I\) mark the number of edges and the degrees of the root and distinguished faces, are respectively \(\widehat M_ g(x,y,z_ I)=\sum_{j \geq 0} y^ j \widehat M^ j_ g(x,z_ I)\), \(\widetilde M_ g (x,y,z_ I)=\sum_{j \geq 0}y^ j \widetilde M^ j_ g (x,z_ I)\). “The planar cases have been studied by E. A. Bender and E. R. Canfield [The number of degree restricted rooted maps on the sphere, SIAM J. Discrete Math. 7, No. 1, 9-15 (1994)]. Using Brown’s result on radicals of a formal power series [On the existence of square roots in certain rings of power series, Math. Ann. 158, 82-89 (1965; Zbl 0136.025); On \(k\)-th roots in power series rings, ibid. 170, 327-333 (1967; Zbl 0144.035)] they derived algebraic equations which \(\widehat M_ 0(x,y,\varnothing)\) satisfies. We extend their results to general surfaces... Asymptotic results ... are also derived for some special sets \(D\)”.