The paper by M. Kontsevich and Yu. Manin focuses on the mathematical aspects of topological quantum field theory and its applications to enumerative problems in algebraic geometry. It introduces and axiomatically treats Gromov–Witten classes, discusses their properties for Fano varieties, and defines Cohomological Field Theories. The authors prove that tree-level theories are determined by their correlation functions. The paper also applies these theories to counting rational curves on del Pezzo surfaces and projective spaces. In the introduction, the authors motivate the study of Gromov–Witten classes by discussing the prediction of numerical characteristics of algebraic curves, particularly genus zero curves, in projective manifolds. They highlight the physical significance of a generating function \(\Phi\) and its analytical properties, which can be uniquely defined. For Calabi–Yau manifolds, \(\Phi\) is conjectured to describe a variation of Hodge structure of the mirror dual manifold. The paper then delves into the formalism of Gromov–Witten classes, which form a collection of linear maps \(I_{g,n,\beta}^V\) satisfying various axioms. These axioms are compiled and explained, providing a geometric intuition behind them. The authors note that the geometric construction of these classes has not been fully established even for simple cases like \(\mathbf{P}^1\). The second section of the paper presents the axiomatic treatment of Gromov–Witten classes, including the setup, definition, and comments on the axioms. The authors also discuss the tensor product of GW-systems and the restricted GW-systems, which are useful for enumerative problems involving incidence conditions stated in terms of algebraic cycles. The third section introduces the first Reconstruction Theorem, which states that if \(H^*(V)\) is generated by \(H^2(V)\), then a tree-level system of GW-classes can be uniquely reconstructed from a finite set of codimension zero basic classes. The proof involves several reduction steps, including quadratic relations and the reconstruction of classes from basic classes. The fourth section explores the potential, associativity relations, and quantum cohomology. It defines a potential \(\Phi\) for a \((A, g)\)-structure and shows how to derive such a structure from a tree-level system of GW-classes. The associativity relations are expressed as a system of quadratic differential equations called WDVV-equations. Overall, the paper provides a comprehensive treatment of Gromov–Witten classes, their properties, and their applications to enumerative geometry, making significant contributions to the field.The paper by M. Kontsevich and Yu. Manin focuses on the mathematical aspects of topological quantum field theory and its applications to enumerative problems in algebraic geometry. It introduces and axiomatically treats Gromov–Witten classes, discusses their properties for Fano varieties, and defines Cohomological Field Theories. The authors prove that tree-level theories are determined by their correlation functions. The paper also applies these theories to counting rational curves on del Pezzo surfaces and projective spaces. In the introduction, the authors motivate the study of Gromov–Witten classes by discussing the prediction of numerical characteristics of algebraic curves, particularly genus zero curves, in projective manifolds. They highlight the physical significance of a generating function \(\Phi\) and its analytical properties, which can be uniquely defined. For Calabi–Yau manifolds, \(\Phi\) is conjectured to describe a variation of Hodge structure of the mirror dual manifold. The paper then delves into the formalism of Gromov–Witten classes, which form a collection of linear maps \(I_{g,n,\beta}^V\) satisfying various axioms. These axioms are compiled and explained, providing a geometric intuition behind them. The authors note that the geometric construction of these classes has not been fully established even for simple cases like \(\mathbf{P}^1\). The second section of the paper presents the axiomatic treatment of Gromov–Witten classes, including the setup, definition, and comments on the axioms. The authors also discuss the tensor product of GW-systems and the restricted GW-systems, which are useful for enumerative problems involving incidence conditions stated in terms of algebraic cycles. The third section introduces the first Reconstruction Theorem, which states that if \(H^*(V)\) is generated by \(H^2(V)\), then a tree-level system of GW-classes can be uniquely reconstructed from a finite set of codimension zero basic classes. The proof involves several reduction steps, including quadratic relations and the reconstruction of classes from basic classes. The fourth section explores the potential, associativity relations, and quantum cohomology. It defines a potential \(\Phi\) for a \((A, g)\)-structure and shows how to derive such a structure from a tree-level system of GW-classes. The associativity relations are expressed as a system of quadratic differential equations called WDVV-equations. Overall, the paper provides a comprehensive treatment of Gromov–Witten classes, their properties, and their applications to enumerative geometry, making significant contributions to the field.