In this talk, I will first introduce the Monge-Ampere eigenvalue problem on general bounded convex domains and related analysis. Then I will discuss the recent resolution, in joint work with Ovidiu Savin, of global smoothness of the eigenfunctions of the Monge-Ampere operator on smooth, bounded and uniformly convex domains in all dimensions. A key ingredient in our analysis is boundary Schauder estimates for certain degenerate Monge-Ampere equations.