Polytopes: general and ordinary

The talk surveys results on the numbers of faces of convex polytopes. A major impediment in the subject has been the lack of examples of polytopes of sufficiently varied combinatorial type. In recent years, however, Bisztriczky introduced "multiplices", which generalize simplices, and "ordinary polytopes", which generalize cyclic polytopes. The talk discusses the combinatorics of these polytopes, and indicates how they might be used to get results for general polytopes.