  We study a broad class of two dimensional gauged linear sigma models (GLSMs)
with off-shell N=(2,2) supersymmetry that flow to nonlinear sigma models
(NLSMs) on noncompact geometries with torsion. These models arise from coupling
chiral, twisted chiral, and semichiral multiplets to known as well as to a new
N=(2,2) vector multiplet, the constrained semichiral vector multiplet (CSVM).
We discuss three kinds of models, corresponding to torsionful deformations of
standard GLSMs realizing Kahler, hyperkahler, and Calabi-Yau manifolds. The
(2,2) supersymmetry guarantees that these spaces are generalized Kahler. Our
analysis of the geometric structure is performed at the classical level, but we
also discuss quantum aspects such as R-symmetry anomalies. We provide an
explicit example of a generalized Kahler structure on the conifold.
