This paper presents a detailed study of the Kodaira–Spencer theory of gravity and exact results for quantum string amplitudes in twisted N = 2 theories with central charge $\hat{c} = 3$. The authors develop techniques to compute higher-loop string amplitudes, focusing on the critical case where $\hat{c} = 3$. A key ingredient is the discovery of an anomaly at every genus, captured by a master anomaly equation. In a particular realization of N = 2 theories, the resulting string field theory is equivalent to a topological theory in six dimensions, the Kodaira–Spencer (KS) theory, which is the closed string analog of the Chern–Simon theory. Using the mirror map, this leads to the computation of the number of holomorphic curves of higher genus in Calabi–Yau manifolds. The paper shows that topological amplitudes can be interpreted as computing corrections to superpotential terms in the effective 4d theory from compactifying superstrings on N = 2 theories. It also discusses relations with c = 1 strings. The paper reviews twisted N = 2 theories, their vacuum geometry, and special geometry of Calabi–Yau 3-folds. It discusses coupling these theories to gravity and the properties of n-point functions. The holomorphic anomaly is derived, showing how it affects partition functions and correlation functions. The paper also discusses the open string version of the theory, including tt* in the open string case and the holomorphic anomaly at one-loop and higher loops. The paper explores what topological amplitudes compute, showing that they relate to the Kodaira–Spencer theory as a string field theory of the B-model. It discusses deformations of complex structure, the BV formalism, and the one-loop computation of the KS theory. The paper also discusses the contribution of holomorphic curves to the topological string amplitudes and the physical implications of topological amplitudes for realistic string models. The paper presents solutions to the anomaly equation and Feynman rules for $F_g$, discusses examples such as orbifolds and the quintic 3-fold, and explores the physical implications of topological amplitudes for realistic string models. It concludes with open problems and prospects for future work. The paper also includes appendices discussing the computation of contributions from bubbling spheres and further analysis of the master anomaly equation.This paper presents a detailed study of the Kodaira–Spencer theory of gravity and exact results for quantum string amplitudes in twisted N = 2 theories with central charge $\hat{c} = 3$. The authors develop techniques to compute higher-loop string amplitudes, focusing on the critical case where $\hat{c} = 3$. A key ingredient is the discovery of an anomaly at every genus, captured by a master anomaly equation. In a particular realization of N = 2 theories, the resulting string field theory is equivalent to a topological theory in six dimensions, the Kodaira–Spencer (KS) theory, which is the closed string analog of the Chern–Simon theory. Using the mirror map, this leads to the computation of the number of holomorphic curves of higher genus in Calabi–Yau manifolds. The paper shows that topological amplitudes can be interpreted as computing corrections to superpotential terms in the effective 4d theory from compactifying superstrings on N = 2 theories. It also discusses relations with c = 1 strings. The paper reviews twisted N = 2 theories, their vacuum geometry, and special geometry of Calabi–Yau 3-folds. It discusses coupling these theories to gravity and the properties of n-point functions. The holomorphic anomaly is derived, showing how it affects partition functions and correlation functions. The paper also discusses the open string version of the theory, including tt* in the open string case and the holomorphic anomaly at one-loop and higher loops. The paper explores what topological amplitudes compute, showing that they relate to the Kodaira–Spencer theory as a string field theory of the B-model. It discusses deformations of complex structure, the BV formalism, and the one-loop computation of the KS theory. The paper also discusses the contribution of holomorphic curves to the topological string amplitudes and the physical implications of topological amplitudes for realistic string models. The paper presents solutions to the anomaly equation and Feynman rules for $F_g$, discusses examples such as orbifolds and the quintic 3-fold, and explores the physical implications of topological amplitudes for realistic string models. It concludes with open problems and prospects for future work. The paper also includes appendices discussing the computation of contributions from bubbling spheres and further analysis of the master anomaly equation.