Résumé : In moduli theory, it happens often that a moduli space is constructed as a quotient. This is a powerful tool, in fact from the quotient structure one can infer several interesting properties about the properties of the moduli space itself. In this talk, I will recall briefly a construction of the moduli stack of trigonal curves as a quotient stack, that I gave in a joint work with Vistoli a few years ago. Then I will move to a recent work with Loenne, where we draw from this construction some surprising results on the fundamental group of the moduli space, that reveals to be of completely different nature from the space of hyperelliptic curve.