The Optimal Transportation problem with quadratic cost can be solved via the elliptic Monge-Ampère Partial Differential Equation (PDE) with nonlocal boundary conditions. Up until now, building numerical solutions for the Monge-Ampère PDE has been a challenge, even with Dirichlet boundary conditions. Indirect methods, such at the fluids reformulation by Brenier-Benamou, as possible, but a direct solver is preferable. Recently several groups of researchers have proposed numerical schemes. Unfortunately these schemes fail to converge, or converge only in the case of smooth solutions. I'll show how naive schemes can work well for smooth solutions, but break down in the singular case. This makes having a convergent scheme even more important. Starting with the Dirichlet problem, I will present a finite difference scheme which is the only scheme proven to converge to weak (viscosity) solutions. Building on the original discretization, I'll describe modifications which improve the accuracy and solution speed. Finally, I will show how to solve the problem with Optimal Transportation boundary conditions. This is joint work with Jean-David Benamou and Brittany Froese.