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For a compact group G, an orbitope is the convex hull of the orbit of a point in some representation of G. For finite groups this construction yields some of the nicest polytopes, among them cubes, demicubes, and permutahedra. In general, orbitopes are compact convex bodies for which the face lattices are much harder to describe. In this talk, I report on some recent results regarding the face lattices of some orbitopes for SO(d). In the cases considered, the facial structure is governed by the combinatorics of polytopes. This is joint work with Frank Sottile and Bernd Sturmfels.