  We prove 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, embedded in a theory with an enlarged gauge
symmetry, with a Landau-Ginzburg theory of Toda type. Standard R -> 1/R duality
and dynamical generation of superpotential by vortices are crucial in the
derivation. This provides not only a proof of mirror symmetry in the case of
(local and global) Calabi-Yau manifolds, but also for sigma models on manifolds
with positive first Chern class, including deformations of the action by
holomorphic isometries.
