Abstract: We study the topological aspects of dessins (via analytic description) with two distinct goals. Firstly we are interested in fields of definition and fields of moduli. We give a topological proof that there exist some dessins with no model defined over their field of moduli. This answers explicitly a question asked in Harbater. Our second motivation is to collect practical and theoretical data for the explicit computation of covers given by some topological description, following ideas of Atkin, Oesterle and ourselves. This leads to a method for the computation of the linear space associated to a divisor on a given dessin.