In this talk I will give an introduction to the cluster algebras of Fomin and Zelevinsky and discuss their categorification via the cluster categories of Buan, Marsh, Reineke, Reiten, and Todorov. I will then talk about the higher cluster categories of Oppermann and Thomas and their relationship with a family of polytopes known as cyclic polytopes in even dimensions. I will finish by mentioning some of my own work in this area, namely showing how to extend this picture to odd-dimensional cyclic polytopes.
