Arithmetic geometry hidden in graphs on surfaces

Several decades ago Grothendieck has established a striking correspondence between certain classes of objects that belong to combinatorial topology (dessins d'enfants) and to arithmetic geometry (Belyi pairs) respectively. A dessin d'enfant is a graph embedded in a surface in such a way that its complement is a disjoint union of open cells; a Belyi pair is a curve over the field of algebraic numbers together with a rational function on it with no more then three critical values.
The talk will be devoted to the problems arising from the above correspondence. In particular, we are going to concentrate on the structures that arise naturally in one category and look mysterious in the other, such as the action of the absolute Galois group of the field of rational numbers on Belyi pairs and the recent (2013) Zograf enumeration of dessins d'enfants.
Some relations with several domains of mathematics and physics will be mentioned and some directions of further research outlined.