A multi-graph\, embedded into a closed connected oriented s
urface\, is calledÂ a 'dessin d'enfant' if its complement is homeomorphic
to a disjoint union of cells. The (appropriately defined) category of des
sins d'enfants turned out to be equivalent to the category of 'Belyi pairs
' that belongs to arithmetic geometry\; a first part of the talk will be d
evoted to the foundations of this theory.

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\nThen the certain prob
lems of enumeration of dessins (and their generalizations) will be introdu
ced. The particular cases of these problems\, e.g.\, recursions for the so
-called 'Hurwitz numbers'\, were studied intensively during the last decad
es\; the corresponding recent results will be mentioned. However\, the gen
eral case is currently out of reach\, and during the second part of the ta
lk a certain project\, based on the above category equivalence\, will be p
resented. Hopefully\, realization of this project will promote the underst
anding of general case.