Calculating and drawing Belyi pairs

The article contains a survey of the current state of the constructive part of the theory of dessin d'enfants. Namely, it is devoted to the actual establishing the correspondence between Belyi pairs and their combinatorial-topological representation. This correspondence is established in terms of the categorical equivalences, for which the necessary categories are introduced. Several connections with arithmetic are discussed. A section is devoted to one of the possible generalizations of the theory, in which the 3 branch points, allowed for the Belyi functions, are replaced by 4. Several direction of further research are presented.