Abstract: In this talk, we discuss the global $C^{2,\alpha}$ estimates for the non-degenerate Monge-Ampere equation in convex polygonal domains in the plane. A crucial ingredient in our analysis is the classification of global solutions to the Monge-Ampere equation in the first quadrant in the plane with constant right hand side and quadratic boundary data. This is joint work with Ovidiu Savin.