In 1781, the French mathematician Gaspard Monge formulated the optimal transportation problem that deals with the relocation of materials in the most economical way. The physical interpretation is to look for an optimal scheme of transporting a pile of soil or rubble to an excavation or fill, which requires the least work. Since then this optimisation problem and its many applications have been extensively studied. In 1940s, the Soviet mathematician and economist Leonid Kantorovich introduced a dual functional, by which the existence of optimal schemes can be established for a large class of cost functions. But for the original cost function proposed by Monge, the proof relies on a very different and more delicate analysis. In this project, you could look into a wide variety of techniques used to construct an optimal scheme for the Monge mass transportation problem. The uniqueness/non-uniqueness, smoothness/non-smoothness phenomenon for optimal schemes, and possible applications in this area will also be discussed.