We give a detailed proof of the bijectivity of the Nash map for normal surface singularities in characteristic 0; this means that the number of irreducible components of the space of arcs on the surface centered at a singular point coincides with the number of irreducible exceptional curves above this point on its minimal resolution of singularities. This proof is extracted from the results of de Fernex Docampo concerning the Nash map for singularities in any dimension.