This paper explores the mathematical foundations of topological quantum field theory and its applications to enumerative geometry, focusing on Gromov-Witten classes and their properties for Fano varieties. It introduces the concept of Cohomological Field Theories (CFTs) and proves that tree-level theories are determined by their correlation functions. The paper discusses applications to counting rational curves on del Pezzo surfaces and projective spaces. The paper begins by introducing Gromov-Witten classes, which are linear maps defined on the cohomology of a projective algebraic manifold. These classes are expected to satisfy a set of formal and geometric properties. The authors use the concept of associativity, or WDVV equations, to show that for Fano manifolds, these equations are strong enough to uniquely define the generating function (potential) $\Phi$ up to a finite number of constants. For Calabi-Yau varieties, these equations are less restrictive. The paper then delves into the geometric interpretation of $\Phi$, showing that it induces a structure of a (super)commutative associative algebra on the tangent bundle of the cohomology space. It also defines a flat connection on this tangent bundle, which is used to show that the associativity equations form a completely integrable system. Additionally, $\Phi$ is shown to induce an extended connection on the tangent bundle lifted to $H^*(V) \times \mathbb{P}^1$, which may define a variation of Hodge structure. The paper discusses the axiomatic treatment of Gromov-Witten classes, including their grading, $S_n$-covariance, and other properties. It also introduces the concept of codimension zero classes, which are particularly important as they express the number of solutions to counting problems. The paper then presents the first Reconstruction Theorem, which shows that Gromov-Witten classes can be recursively calculated in certain situations. The paper also discusses the construction of Cohomological Field Theories, which are less constrained structures that focus on moduli spaces rather than the target manifold. It introduces the concept of tensor product of GW-systems and cusp classes, which are elements of the cohomology of moduli spaces that vanish on all boundary divisors. Finally, the paper presents the second Reconstruction Theorem, which allows the classification of Cohomology Field Theories via solutions of WDVV equations and the formal proof of the existence of GW-classes for projective spaces. The paper concludes by emphasizing the importance of these results in the context of quantum field theory and their implications for enumerative geometry.This paper explores the mathematical foundations of topological quantum field theory and its applications to enumerative geometry, focusing on Gromov-Witten classes and their properties for Fano varieties. It introduces the concept of Cohomological Field Theories (CFTs) and proves that tree-level theories are determined by their correlation functions. The paper discusses applications to counting rational curves on del Pezzo surfaces and projective spaces. The paper begins by introducing Gromov-Witten classes, which are linear maps defined on the cohomology of a projective algebraic manifold. These classes are expected to satisfy a set of formal and geometric properties. The authors use the concept of associativity, or WDVV equations, to show that for Fano manifolds, these equations are strong enough to uniquely define the generating function (potential) $\Phi$ up to a finite number of constants. For Calabi-Yau varieties, these equations are less restrictive. The paper then delves into the geometric interpretation of $\Phi$, showing that it induces a structure of a (super)commutative associative algebra on the tangent bundle of the cohomology space. It also defines a flat connection on this tangent bundle, which is used to show that the associativity equations form a completely integrable system. Additionally, $\Phi$ is shown to induce an extended connection on the tangent bundle lifted to $H^*(V) \times \mathbb{P}^1$, which may define a variation of Hodge structure. The paper discusses the axiomatic treatment of Gromov-Witten classes, including their grading, $S_n$-covariance, and other properties. It also introduces the concept of codimension zero classes, which are particularly important as they express the number of solutions to counting problems. The paper then presents the first Reconstruction Theorem, which shows that Gromov-Witten classes can be recursively calculated in certain situations. The paper also discusses the construction of Cohomological Field Theories, which are less constrained structures that focus on moduli spaces rather than the target manifold. It introduces the concept of tensor product of GW-systems and cusp classes, which are elements of the cohomology of moduli spaces that vanish on all boundary divisors. Finally, the paper presents the second Reconstruction Theorem, which allows the classification of Cohomology Field Theories via solutions of WDVV equations and the formal proof of the existence of GW-classes for projective spaces. The paper concludes by emphasizing the importance of these results in the context of quantum field theory and their implications for enumerative geometry.