This paper presents a proof of mirror symmetry for supersymmetric sigma models on Kahler manifolds in 1+1 dimensions. The proof involves establishing the equivalence of the gauged linear sigma model with a Landau-Ginzburg theory of Toda type. Key elements of the proof include the R → 1/R duality and the dynamical generation of superpotential by vortices. The result applies not only to Calabi-Yau manifolds but also to sigma models on manifolds with positive first Chern class, including deformations by holomorphic isometries. Mirror symmetry is a duality that exchanges complex deformations on one manifold with Kahler deformations on the dual manifold. It has been supported by many examples and has led to significant progress in understanding non-perturbative effects in string theory. The paper also discusses the extension of mirror symmetry to more general cases, including sigma models with positive first Chern class and non-compact Calabi-Yau manifolds. The proof of mirror symmetry relies on establishing a dual description of (2,2) supersymmetric gauge theories in 1+1 dimensions. This involves dualizing the phase of charged fields using R → 1/R duality and describing the low energy effective theory in terms of dual variables. A superpotential is dynamically generated by the instanton effect, and the (twisted) F-term part of the effective Lagrangian is exactly determined. The paper relates these results to the statement of mirror symmetry by connecting U(1) gauge theories with matter to sigma models on Kahler manifolds. It shows that the mirror of a sigma model is a Landau-Ginzburg model. In the case of Calabi-Yau manifolds, this can also be related to the sigma model on another Calabi-Yau manifold via the equivalence of sigma models and Landau-Ginzburg models. For manifolds with non-zero first Chern class, the axial U(1) R-symmetry is broken by an anomaly, and the vector U(1) R-symmetry of the mirror theory must be broken by an inhomogeneous superpotential. The paper also discusses the mirror symmetry of the CP^1 sigma model, which is mirror to the N=2 sine-Gordon theory. The linear sigma model for CP^1 is a U(1) gauge theory with two chiral multiplets of charge Q1=Q2=1. By integrating out the Σ field, the constraint Y1 + Y2 = t is obtained, which can be solved by Y1 = Y + t/2, Y2 = -Y + t/2. The superpotential then takes the form of the sine-Gordon potential. The paper concludes by discussing the implications of these results for the classification of vacua and the behavior of the theory in the infrared limit. It also outlines the organization of the paper, which includes reviews of mirror symmetry, theThis paper presents a proof of mirror symmetry for supersymmetric sigma models on Kahler manifolds in 1+1 dimensions. The proof involves establishing the equivalence of the gauged linear sigma model with a Landau-Ginzburg theory of Toda type. Key elements of the proof include the R → 1/R duality and the dynamical generation of superpotential by vortices. The result applies not only to Calabi-Yau manifolds but also to sigma models on manifolds with positive first Chern class, including deformations by holomorphic isometries. Mirror symmetry is a duality that exchanges complex deformations on one manifold with Kahler deformations on the dual manifold. It has been supported by many examples and has led to significant progress in understanding non-perturbative effects in string theory. The paper also discusses the extension of mirror symmetry to more general cases, including sigma models with positive first Chern class and non-compact Calabi-Yau manifolds. The proof of mirror symmetry relies on establishing a dual description of (2,2) supersymmetric gauge theories in 1+1 dimensions. This involves dualizing the phase of charged fields using R → 1/R duality and describing the low energy effective theory in terms of dual variables. A superpotential is dynamically generated by the instanton effect, and the (twisted) F-term part of the effective Lagrangian is exactly determined. The paper relates these results to the statement of mirror symmetry by connecting U(1) gauge theories with matter to sigma models on Kahler manifolds. It shows that the mirror of a sigma model is a Landau-Ginzburg model. In the case of Calabi-Yau manifolds, this can also be related to the sigma model on another Calabi-Yau manifold via the equivalence of sigma models and Landau-Ginzburg models. For manifolds with non-zero first Chern class, the axial U(1) R-symmetry is broken by an anomaly, and the vector U(1) R-symmetry of the mirror theory must be broken by an inhomogeneous superpotential. The paper also discusses the mirror symmetry of the CP^1 sigma model, which is mirror to the N=2 sine-Gordon theory. The linear sigma model for CP^1 is a U(1) gauge theory with two chiral multiplets of charge Q1=Q2=1. By integrating out the Σ field, the constraint Y1 + Y2 = t is obtained, which can be solved by Y1 = Y + t/2, Y2 = -Y + t/2. The superpotential then takes the form of the sine-Gordon potential. The paper concludes by discussing the implications of these results for the classification of vacua and the behavior of the theory in the infrared limit. It also outlines the organization of the paper, which includes reviews of mirror symmetry, the