The theory of dessin d’enfants was started out by on observation of Groethendieck in Esquisse d’un programme, 1984. Loosely speaking, a dessin d’enfants consists of a pair (X,D), where X is an orientable topological surface and D is a connected 1-complex D subset X, such that X – D consists of open cells. To such a combinatorial data, Groethendieck observed that one can associate, a smooth algebraic curve X_D together with a map X_D to P^1, ramified only over three points, and everything will be defined over the algebraic closure of Q thank’s to Belyi’s Theorem. Moreover, one can also go the other way around: to every smooth, algebraic curve C with a rational map ramified only over three points, one can associate a dessin d’enfants, and the two constructions are equivalent. In this talk, we will explain this construction and describe some examples coming from Shabat and Voevodsky’s paper “Drawing curves over number field”. For some particular dessins, we will show how to explicitly construct a Fucsian group Gamma, such that H / Gamma cong X_D. In the last part of the talk, we will introduce the Galois action on the set of dessins, proving that it is faithful for genus 0 and 1 dessins if time permits.