The Chow ring of the stack of stable curves of genus 2

There is by now an extensive theory of rational Chow rings of moduli spaces of smooth curves. There is an integral version of these Chow rings: the theory is well understood, but calculations are difficult, and in concrete cases have been carried out only in sporadic cases. I will survey the little that is known, and discuss recent joint work with Andrea Di Lorenzo on stable curves of genus 2.