We discuss recent a priori interior Hessian estimates for solutions of the special Lagrangian equation, when the equation has phase at least a certain value, or when the solution is convex. These equations include the sigma-2 equation in dimension three. The gradient graph of any solution is a minimizing Lagrangian surface. While Heinz showed in the 50's that similar estimates hold for the sigma-2 (Monge-Ampere) equation in dimension two, Pogorelov showed that such estimates cannot hold for the sigma-3 (Monge-Ampere) equation in dimension three. This is joint work with Yu Yuan, partly also with Jingyi Chen.