**Summary:** This paper by David Mumford explores the enumerative geometry of the moduli space of curves, focusing on defining a Chow ring for the moduli space $ \overline{M}_g $ of stable curves and its compactification. The goal is to establish a framework for studying geometrically important loci in $ \overline{M}_g $ using a set of "tautological" classes $ \kappa_i $, which are defined in terms of Chern classes of universal bundles. The paper also addresses the challenges posed by the singular nature of $ \overline{M}_g $, which is not unirational for large $ g $, and introduces techniques from Fulton and MacPherson to define a Chow ring for such spaces. The first part of the paper defines a Chow ring for $ \overline{M}_g $ by using a resolution of singularities and the Grothendieck-Riemann-Roch theorem. It introduces the concept of Q-varieties, which are varieties that are locally quotients of smooth varieties by finite groups, and discusses their properties. The second part introduces the tautological classes $ \kappa_i $ and explores their relationships, as well as the fundamental classes of certain subvarieties like the hyperelliptic locus. The third part provides a concrete example by computing the Chow ring for $ \overline{M}_2 $, showing that $ M_2 $ is affine. The paper also discusses the use of Q-stacks for moduli spaces like $ \overline{M}_2 $ and $ \overline{M}_{1,1} $, where the general object has automorphisms. It introduces the concept of a Q-stack and shows how it can be used to define a Chow ring for such spaces. The paper concludes by establishing a key result that relates the Chow group of a Q-variety with the G-invariants of the operational Chow ring of its normalization, under the assumption that the normalization is Cohen-Macaulay. This result is crucial for defining a ring structure on the Chow group of the Q-variety and for computing Chern classes of coherent sheaves on it. The paper also discusses the implications of this result for the study of the Chow ring of Q-varieties and Q-stacks, emphasizing the importance of rational equivalence and the role of the moving lemma in ensuring the well-definedness of the intersection product.**Summary:** This paper by David Mumford explores the enumerative geometry of the moduli space of curves, focusing on defining a Chow ring for the moduli space $ \overline{M}_g $ of stable curves and its compactification. The goal is to establish a framework for studying geometrically important loci in $ \overline{M}_g $ using a set of "tautological" classes $ \kappa_i $, which are defined in terms of Chern classes of universal bundles. The paper also addresses the challenges posed by the singular nature of $ \overline{M}_g $, which is not unirational for large $ g $, and introduces techniques from Fulton and MacPherson to define a Chow ring for such spaces. The first part of the paper defines a Chow ring for $ \overline{M}_g $ by using a resolution of singularities and the Grothendieck-Riemann-Roch theorem. It introduces the concept of Q-varieties, which are varieties that are locally quotients of smooth varieties by finite groups, and discusses their properties. The second part introduces the tautological classes $ \kappa_i $ and explores their relationships, as well as the fundamental classes of certain subvarieties like the hyperelliptic locus. The third part provides a concrete example by computing the Chow ring for $ \overline{M}_2 $, showing that $ M_2 $ is affine. The paper also discusses the use of Q-stacks for moduli spaces like $ \overline{M}_2 $ and $ \overline{M}_{1,1} $, where the general object has automorphisms. It introduces the concept of a Q-stack and shows how it can be used to define a Chow ring for such spaces. The paper concludes by establishing a key result that relates the Chow group of a Q-variety with the G-invariants of the operational Chow ring of its normalization, under the assumption that the normalization is Cohen-Macaulay. This result is crucial for defining a ring structure on the Chow group of the Q-variety and for computing Chern classes of coherent sheaves on it. The paper also discusses the implications of this result for the study of the Chow ring of Q-varieties and Q-stacks, emphasizing the importance of rational equivalence and the role of the moving lemma in ensuring the well-definedness of the intersection product.