Boundary estimates for second derivatives of solutions to the Dirichlet problem for the Monge-Ampere equation were first obtained by Ivockina in 1980. A few years later independently Krylov and Caffarelli-Nirenberg-Spruck obtained $C^{2,\alpha}$ regularity of solutions and this led to the solvability of the classical Dirichlet problem in the case when the data is sufficiently smooth. In our talk we will discuss some more recent results about boundary regularity under optimal assumptions on the data.