Abstract : This thesis deals with Grothendieck-Teichmüller theory and moduli spaces of curves with un- ordered marked points, more specially with different types of inertia in the corresponding geometric fundamental groups. We extend the known absolute Galois action on divisorial inertia at infinity to an analogous action on stack inertia in genus zero, and on all prime order profinite torsion elements in genus zero and one. In fact, we show that the latter result holds not only for the ab solute Galois group but for a new version GS of the Grothendieck-Teichmüller group coming from torsion conditions in genus zero, which as we show, acts on full profinite mapping class groups in all genera. We show this result by adapting a cohomological principle due to J. P. Serre which reduces torsion of a profinite group to discrete torsion in certain cases. We use this theory to show that prime order profinite torsion in the genus zero and one mapping class groups is conjugate to discrete torsion. We then use this to determine the action of GS on prime order profinite torsion. Finally, we show in genus zero how to obtain the absolute Galois group action on stack inertia by using geometry of special loci.