Moduli spaces of curves and their etale fundamental group representations have a long tradition in the study of the absolute Galois group of rationals. That led in particular to the creation of Grothendieck-Teichmuller theory, which in return provides a group theoretic approach to arithmetic questions. The goal of this talk is to present how the GT and arithmetic approaches lead to new arithmetic results in the study of the stack stratification of the space such that given by the automorphism group of curves. I will first show how Grothendieck-Teichmuller theory, in relation with explicit presentations and fundamental properties of the mapping class group of surfaces, gives a result for the first cyclic stack strata in low genus, then explain how the switch to a geometric context extends this result to the generic cyclic strata in every genus. My presentation will also illustrate how this stack arithmetic is organized around two essential arithmetic questions, that are for the moduli spaces to be etale K(\pi,1) and the rationality of irreducible components in Hurwitz spaces.