  This paper intents to present the state of art and recent developments of the
optimal transportation theory with many marginals for a class of repulsive cost
functions. We introduce some aspects of the Density Functional Theory (DFT)
from a mathematical point of view, and revisit the theory of optimal transport
from its perspective. Moreover, in the last three sections, we describe some
recent and new theoretical and numerical results obtained for the Coulomb cost,
the repulsive harmonic cost and the determinant cost.
