The issue of regularity of optimal transport maps in the case ``cost$\,=\,$squared distance'' on $\mathbb R^n$ was solved by Caffarelli in the 1990s. However, a major open problem in the theory was the question of regularity for more general cost functions, or for the case ``cost$\,=\,$squared distance'' on a Riemannian manifold. A breakthrough to this problem has been achieved by Ma-Trudinger-Wang (2005) and Loeper (2007), who found a necessary and sufficient condition on the cost function in order to ensure the regularity of the optimal map. This condition, now called MTW condition, involves a combination of derivatives of the cost, up to the fourth order. In the special case ``cost$\,=\,$squared distance'' on a Riemannian manifold, the MTW condition corresponds to ask for the non-negativity of a new curvature tensor on the manifold (the so-called MTW tensor), and it implies strong geometric consequences on the geometry of the manifold and on the structure of its cut-locus.