In this talk, we consider an optimal transport problem arising from stopping the Brownian motion from a given distribution to get a fixed or free target distribution. We explain that under certain natural assumptions on the transportation cost, the optimal stopping time is given by the hitting time to a barrier, which is determined by the solution to the dual optimization problem. In the free target case, the problem is related to the Stefan problem, that is, a free boundary problem for the heat equation. We obtain analytical information on the optimal solutions, including certain BV estimates. The fixed target case is mainly from the joint work with Nassif Ghoussoub and Aaron Palmer at UBC, while the free target case is the recent joint work-in-progress with Inwon Kim at UCLA.