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A dessin d'enfant, or dessin, is a bicolored graph embedded into a Riemannsurface, and the monodromy group is an algebraic invariant of the dessingenerated by rotations of edges about black and white vertices. A rationalpolygonal billiards surface is a Riemann surface that arises from the dynamicalsystem of billiards within a rational-angled polygon. In this paper, we computethe monodromy groups of dessins embedded into rational polygonal billiardssurfaces and identify all possible monodromy groups arising from rationaltriangular billiards surfaces.

Monodromy Groups of Dessins d'Enfant on Rational Polygonal Billiards Surfaces