There is by now an extensive theory of rational Chow rings of moduli spaces of smooth curves. The integral version of these Chow rings is not as well understood. I will survey what is known. In the last part of the talk I will discuss the Chow ring of the stack of stable curves of genus 2, which has been recently calculated by Eric Larson. I will present a different approach to the calculation, which offers an interesting point of view on stack of stable curves of genus 2. This part is joint work with Andrea Di Lorenzo.