In this talk, I will present some recent results on the regularity of free boundaries in optimal transportation. In particular, for higher order regularity, when the densities are $\alpha$-Hölder and the domains are $C^2$, uniformaly convex, we obtain the free boundary is $C^{2,\alpha}$. We also study a model case where the target consists of two disjoint convex sets, in which singularities of the optimal transport mapping arise. Under suitable assumptions, we prove that the singular set of the optimal mapping is an $(n-1)$-dimensional $C^{2,\alpha}$ regular sub-manifold of $\mathbb{R}^n$. These results are based on a series of joint works with Shibing Chen and Xu-Jia Wang.