Moduli of curves play a prominent role in algebraic geometry. In particular, their rational Chow rings have been the subject of intensive research in the last forty years, since Mumford first investigated the subject. There is also a well defined notion of integral Chow ring for these objects: this is more refined, but also much harder to compute. In this talk I will present the computation of the integral Chow ring of moduli of stable 1-pointed curves of genus two, obtained by using a new approach to this type of questions (joint work with Michele Pernice and Angelo Vistoli).