  This contribution summarizes recent work of the authors that combines methods
from dynamical systems theory (discrete Painlev\'e equations) and asymptotic
analysis of orthogonal polynomial recurrences, to address long-standing
questions in map enumeration. Given a genus $g$, we present a framework that
provides the generating function for the number of maps that can be realized on
a surface of that genus. In the case of 4-valent maps, our methodology leads to
explicit expressions for map counts. For general even or mixed valence, the
number of vertices of the map specifies the relevant order of the derivatives
of the generating function that needs to be considered. Beyond summarizing our
own results, we provide context for the program highlighted in this article
through a brief review of the literature describing advances in map
enumeration. In addition, we discuss open problems and challenges related to
this fascinating area of research that stands at the intersection of
statistical physics, random matrices, orthogonal polynomials, and discrete
dynamical systems theory.
