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The main purpose of this thesis is to study computational questions related to compact Riemann surfaces (algebraic curves) with main emphasis on the theory of Grothendieck’s dessins d’enfants; where recovering the explicit formula of a Belyi map from the corresponding dessin on a topological surface is important. Some applications such as investigating the modular j-function and also certain classes of modular curves are included in this thesis along with a summary of attempts toward generalizing the Belyi theorem to complex dimension two.First two chapters contain necessary prerequisites on compact Riemann surfaces and a short introduction to the theory of dessins d’enfants and Belyi...