TITLE: Generalizations of the Birkhoff-von Neumann Theorem
QUESTION [31 upvotes]: 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: Sampling from the Birkhoff polytope

REPLY [3 votes]: The theorem still holds if we ask each row and column to sum to some integer $m$; permutation matrices are replaced with zero-one matrices having the same constraints. See Watkins and Merris, Convex Sets of Doubly Stochastic Matrices and Lewandowski, Liu and Liu, An Algorithmic Proof of a Generalization of the
Birkhoff - Von Neumann Theorem.