  In the first part of the paper we briefly decribe the classical problem,
raised by Monge in 1781, of optimal transportation of mass. We discuss also
Kantorovich's weak solution of the problem, which leads to general existence
results, to a dual formulation, and to necessary and sufficient optimality
conditions. In the second part we describe some recent progress on the problem
of the existence of optimal transport maps. We show that in several cases
optimal transport maps can be obtained by a singular perturbation technique
based on the theory of $\Gamma$-convergence, which yields as a byproduct
existence and stability results for classical Monge solutions.
