We want to present some new results related to Multi-marginal Optimal Transport Theory for Coulomb and repulsive harmonic costs. In particular, we are inquiring about the existence of Monge-type solutions in the multi-marginal case. We intend to give a brief survey on the general theory and discuss some of its main issues. Most of the results we will present in this talk are contained in a joint work with S. Di Marino and L. Nenna.