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Preface Chapter 0.Introduction: What Is Quantum Cohomology? Chapter I.Introduction to Frobenius Manifolds 1.Definition of Frobenius manifolds and the structure connection 2.Identity, Euler field, and the extended structure connection 3.Semisimple Frobenius manifolds 4.Examples 5.Weak Frobenius manifolds Chapter II.Frobenius Manifolds and Isomonodromic Deformations 1.The second structure connection 2.Isomonodromic deformations 3.Semisimple Frobenius manifolds as special solutions to the Schlesinger equations 4.Quantum cohomology of projective spaces 5.Dimension three and Painleve VI Chapter III.Frobenius Manifolds and Moduli Spaces of Curves 1.Formal Frobenius manifolds and Commoo-algebras 2.Pointed curves and their graphs 3.Moduli spaces of genus 0 4.Formal Frobenius manifolds and Cohomological Field Theories 5.Gromov-Witten invariants and quantum cohomology: Axiomatic theory 6.Formal Frobenius manifolds of rank one and Weil-Petersson volumes of moduli spaces 7.Tensor product of analytic Frobenius manifolds 8.K.Salto's frameworks and singularities 9.Maurer-Cartan equations and Gerstenhaber-Batalin-Vilkovyski algebras 10.From dGBV-algebras to Frobenius manifolds Chapter IV.Operads, Graphs, and Perturbation Series 1.Classical linear opera(is 2.Operads and graphs 3.Sums over graphs 4.Generating functions Chapter V.Stable Maps, Stacks, and Chow Groups 1.Prestable curves and prestable maps 2.Flat families of curves and maps 3.Groupoids and moduli groupoids 4.Morphisms of groupoids and moduli groupoids 5.Stacks 6.Homological Chow groups of schemes 7.Homological Chow groups of DM--stacks 8.Operational Chow groups of schemes and DM-stacks Chapter VI.Algebraic Geometric introduction to the Gravitational Quantum Cohomology 1.Virtual fundamental classes 2.Gravitational descendants and Virasoro constraints 3.Correlators and forgetful maps 4.Correlators and boundary maps 5.The simplest Virasoro constraints 6.Generalized correlators 7.Generating functions on the large phase space Bibliography Subject Index