  We use the gauged linear sigma model introduced by Witten to calculate
instanton expansions for correlation functions in topological sigma models with
target space a toric variety $V$ or a Calabi--Yau hypersurface $M \subset V$.
In the linear model the instanton moduli spaces are relatively simple objects
and the correlators are explicitly computable; moreover, the instantons can be
summed, leading to explicit solutions for both kinds of models. In the case of
smooth $V$, our results reproduce and clarify an algebraic solution of the $V$
model due to Batyrev. In addition, we find an algebraic relation determining
the solution for $M$ in terms of that for $V$. Finally, we propose a
modification of the linear model which computes instanton expansions about any
limiting point in the moduli space. In the smooth case this leads to a (second)
algebraic solution of the $M$ model. We use this description to prove some
conjectures about mirror symmetry, including the previously conjectured
``monomial-divisor mirror map'' of Aspinwall, Greene, and Morrison.
