In this talk, I will introduce the Monge-Ampere eigenvalue problem and discuss the global smoothness of the eigenfunctions. The question of global higher derivative estimates up to the boundary of the eigenfunctions of the Monge-Ampere operator is a well known open problem. I will discuss the recent resolution 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. This is joint work with Ovidiu Savin.