A prerequisite for understanding this paper, as the authors admit, is a book by the third author [Enumerative theory of maps (Mathematics and its Applications 468, Kluwer, Dordrecht/Boston/London) (1999; Zbl 0990.05070)]. Lacking ready access to this book, the reviewer is reduced to quoting the abstract of this paper: This paper provides some functional equations and parametric experssions of \(g\)-essential maps on the projective plane, on the torus and on the Klein bottle with the size as a parameter and gives their explicit formulae for exact enumeration further.