Given a projective variety X of dimension d, a "flag" of subvarieties Y_i, and a big divisor D, Okounkov showed how to construct a convex body in R^d, and this construction has recently been developed further in work of Kaveh-Khovanskii and Lazarsfeld-Mustata. In general, this Okounkov body is quite hard to understand, but when X is a toric variety, it is just the polytope associated to D via the standard yoga of toric geometry. I´ll describe a more general situation where the Okounkov body is still a polytope, and show that in this case X admits a flat degeneration to the corresponding toric variety. This project was motivated by examples, and as an application, I´ll describe some toric degenerations of flag varieties and Schubert varieties. There will be pictures of polytopes.