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The Landau-Ginzburg/Calabi-Yau(LG/CY)correspondence is a proposed equivalence between two enumerative theories associated to a homogeneous polynomial: the Gromov-Witten theory of the hypersurface cut out by the polynomial in projective space,and the Landau-Ginzburg theory of the polynomial when viewed as a singularity.Such a correspondence was originally suggested by Witten in 1993 as part of a far-reaching conjecture relating the "gauged linear sigma models" arising at different phases of a GIT quotient.I will discuss an explicit formulation and proof of Wittens proposal for complete intersections in weighted projective space,generalizing the LG/CY correspondence for hypersurfaces and introducing a number of new features.This is joint work with Dustin Ross.