In this talk I will survey recent advances on the understanding of the combinatorics and geometry of these polyhedra. In particular, I will report on the following ``universality theorem'': Every rational convex polytope is strongly isomorphic to a 3-way, 2-margin transportation polytope. This has very interesting consequences for integer programming and statistics. It is indeed useful to the solution of several open questions collected in the 1984 monograph by Yemelichev-Kovalev-Kravtsov and the 1986 survey paper of Vlach. For young people I will try to state several open questions, even for the classical 2-way case.
Based on joint papers with Ed Kim, Fu Liu, Francisco Santos and Shmuel Onn.