  Delzant's theorem for symplectic toric manifolds says that there is a
one-to-one correspondence between certain convex polytopes in $\mathbb{R}^n$
and symplectic toric $2n$-manifolds, realized by the image of the moment map. I
review proofs of this theorem and the convexity theorem of
Atiyah-Guillemin-Sternberg on which it relies. Then, I describe Honda's results
on the local structure of near-symplectic 4-manifolds, and inspired by recent
work of Gay-Symington, I describe a generalization of Delzant's theorem to
near-symplectic toric 4-manifolds. One interesting feature of the
generalization is the failure of convexity, which I discuss in detail. The
first three chapters are primarily expository, duplicate material found
elsewhere, and may be skipped by anyone familiar with the material, but are
included for completeness.
