I will begin with an introduction to symplectic geometry, and then introduce completely integrable systems arising from torus actions and use this to determine topological information about a symplectic manifold. I will then discuss how we can consider toric varieties as examples of these and how their corresponding (Delzant) polytopes arise naturally in symplectic geometry as images of a "moment map". If time permits I'll talk a bit about how the combinatorics of the fan can be used to calculate even more topological information about a toric variety.