In the theory of toric varieties, the combinatorics of a convex integral

polytope Delta is intimately linked with the geometry of an associated toric variety X(Delta); indeed, the polytope Delta fully encodes the geometry of X(Delta) in this case. In more general situations arising in geometry, one can often associate combinatorial data to a group action on a manifold, but usually the combinatorics doesn’t completely encode the original geometric data. The recent theory of Newton–Okounkov bodies initiated by Kaveh–Khovanskii and Lazarsfeld–Mustata can be viewed as a generalization of the theory of toric varieties to a much more general setting: given an arbitrary algebraic variety X, together with some auxiliary data, this theory produces a convex body Delta, and in many cases Delta is a rational polytope. This talk will be a basic introduction to this relatively recent theory, and some of the questions it raises.