Toric symplectic manifolds are a special subset of symplectic manifolds classified by a nice set of polytopes in $\mathbb{R}^n$. I'll give an introduction to symplectic and toric symplectic manifolds, and then talk about a generalization of symplectic manifolds called origami manifolds. Toric origami manifolds are classified by "origami templates," which in 2 dimensions resemble the folded paper art for which they are named.