Three facets of dessins d'enfants theory (the 2nd day of the Conference devoted to the memory of Gena Belyi on the occasion of his 70th birthday)

As everybody knows, the 3-points theorem proved by Belyi in 1978 enriched Grothendieck's (with his students') considerations of embeddings of graphs into surfaces; that was the Big Bang of dessins d'enfants theory.
I am going to give an overview of the development of this theory during the recent decades, concentrating on the interactions between the algebro-geometric, combinatorial and group-theoretical viewpoints.