Optimal mass transportation: from Kantorovich to Monge, from two to many marginals.

Once upon a time there was a French mathematician called Gaspard Monge who set out to explore the problem of transporting mass from one place to another place in an optimal way.  More than 100 years later, a Russian mathematician called Leonid Kantorovich studied the duality between minimizing the cost and maximizing the benefits of the transport. Today we study 'the Monge problem', 'the Kantorovich Duality', and 'The Monge-Kantorovich problem', named in honor of the two founding fathers of the field. In the most classical formulation of the problem, we move mass from one place (formally: from one 'marginal' measure) to another one, and the transporting gets more expensive when the transportation distance increases. But what happens if we have more than two marginals? What changes if the cost function is repulsive, i.e. increases when the distance of the points to be coupled decreases? Why, in particular, does the Monge problem become so difficult when we move from two to many marginals? And what is this Monge problem in the first place ?