Toric symplectic manifolds are spaces of particular interest to symplectic geometers because their complicated geometry is completely classified by special polytopes that are easy to work with and visualize. I'll introduce symplectic and toric symplectic manifolds, and then talk about a generalization of toric symplectic manifolds called toric origami manifolds. Toric origami manifolds are classified by "origami templates," which in 2 dimensions resemble the folded paper art for which they are named.