In the second part of this seminar we will generalize the result obtained for the moduli space T_5 of trigonal curves of genus 5, to higher genera. Starting from the approach used to compute the rational cohomology of T_5, we will give a description of the rational cohomology ring of T_g in a certain range. We will prove that it is independent of the genus g when it is sufficiently large with respect to the degree and that it coincides with the tautological ring in this stable range.