This thesis studies the counting functions for metric ribbon graphs. Ribbon graphs, also known as combinatorial maps, are cellular embeddings of graphs in surfaces. Metric on a ribbon graph is an assignment of positive lengths to its edges. The counting functions give the number of integral metric ribbon graphs with fixed combinatorics (genus of the surface, degrees of vertices, number of boundaries) as a function of the perimeters of the boundaries. The motivation for their study comes from the geometry of surfaces and their moduli spaces.
Our approach to their study is purely combinatorial and relies on bijections and surgeries for ribbon graphs. We present several polynomiality results, as well as explicit formulas, for the top-degree terms of these functions. These results are then applied to the problem of asymptotic enumeration of square-tiled surfaces according to their number of cylinders. This gives a refinement of several results previously obtained via algebraic geometry.