In this talk, we will discuss some recent regularity results of free boundary in optimal transportation. In particular, we show the free boundary touch the fixed boundary in a nice way, namely no cusp exists, which leads to a global regularity of the free boundary up to the fixed boundary. Additionally, we will consider another model case that the target consists of two disjoint convex sets, in which singularities of optimal transport mapping arise. Under some assumptions, we can obtain regularities of this singular set.
