• Hurwitz problem
• equivalence between covers
• branched covers
• Mednykh formula

This thesis deals with the realizability problem of branched coverings between surfaces proposed by A. Hurwitz in 1891.
We give some different definitions of equivalence between coverings which satisfies a branching datum and we investigate the relations between them making use of two tecniques: the dessins d'enfant and the constellations.
We follows an article of A. Mednykh to give a formula for the inequivalent branched coverings
satisfying a branching datum, proposing an implementation of such a formula throught the platform Mathematica and finally giving an explicit computation for some particular data.