The paper aims to explore the enumerative geometry of the moduli space of curves of arbitrary genus \( g \). The main goal is to define a Chow ring for the moduli space \( \mathcal{M}_g \) and its compactification \( \overline{\mathcal{M}}_g \), and to calculate the classes of certain geometrically important loci in terms of these classes. The author uses the Grassmannian as a model, where the basic classes are the Chern classes of the tautological bundle and the fundamental cycles are the Schubert cycles. The paper is divided into three parts. The first part defines an intersection product in the Chow group of \( \overline{\mathcal{M}}_g \). Due to the singularities of \( \overline{\mathcal{M}}_g \), which are mild, the author uses the fact that \( \overline{\mathcal{M}}_g \) is globally the quotient of a Cohen-Macaulay variety by a finite group, along with ideas from Fulton and MacPherson, and the Riemann-Roch theorem, to define the intersection product. The second part introduces a sequence of "tautological" classes \( \kappa_i \) in \( A^i(\overline{M}_g) \otimes \mathbb{Q} \), derives relations between them, and calculates the fundamental class of certain subvarieties, such as the hyperelliptic locus, in terms of these classes. The Grothendieck Riemann-Roch theorem is a key tool in this part. The third part works out \( A^i(\overline{M}_2) \) completely, providing an example of the theory. An interesting corollary is that \( \mathcal{M}_2 \) is affine. The author suggests that it would be valuable to work out \( A^i(\overline{M}_g) \) or \( H^i(\mathcal{M}_g) \) for other small values of \( g \) to better understand the properties of these rings and their relation to the geometry of \( \overline{M}_g \). Techniques from Atiyah-Bott may be useful in this context.The paper aims to explore the enumerative geometry of the moduli space of curves of arbitrary genus \( g \). The main goal is to define a Chow ring for the moduli space \( \mathcal{M}_g \) and its compactification \( \overline{\mathcal{M}}_g \), and to calculate the classes of certain geometrically important loci in terms of these classes. The author uses the Grassmannian as a model, where the basic classes are the Chern classes of the tautological bundle and the fundamental cycles are the Schubert cycles. The paper is divided into three parts. The first part defines an intersection product in the Chow group of \( \overline{\mathcal{M}}_g \). Due to the singularities of \( \overline{\mathcal{M}}_g \), which are mild, the author uses the fact that \( \overline{\mathcal{M}}_g \) is globally the quotient of a Cohen-Macaulay variety by a finite group, along with ideas from Fulton and MacPherson, and the Riemann-Roch theorem, to define the intersection product. The second part introduces a sequence of "tautological" classes \( \kappa_i \) in \( A^i(\overline{M}_g) \otimes \mathbb{Q} \), derives relations between them, and calculates the fundamental class of certain subvarieties, such as the hyperelliptic locus, in terms of these classes. The Grothendieck Riemann-Roch theorem is a key tool in this part. The third part works out \( A^i(\overline{M}_2) \) completely, providing an example of the theory. An interesting corollary is that \( \mathcal{M}_2 \) is affine. The author suggests that it would be valuable to work out \( A^i(\overline{M}_g) \) or \( H^i(\mathcal{M}_g) \) for other small values of \( g \) to better understand the properties of these rings and their relation to the geometry of \( \overline{M}_g \). Techniques from Atiyah-Bott may be useful in this context.