The famous Birkhoff-von Neumann theorem asserts that every doubly stochastic matrix can be written as a convex combination of permutation matrices. 

The question is to point out different generalizations of this theorem, different "non-generalizations" namely cases where an expected generalization is false, and to briefly describe the context of these generalizations. 

A related MO question: http://mathoverflow.net/questions/73805/sampling-from-the-birkhoff-polytope