The degenerate special Lagrangian (DSL) equation is a fully-nonlinear degenerate elliptic PDE governing the existence of geodesics in the space of positive Lagrangians. Existence of geodesics is a key step in Solomon’s program for the existence of special Lagrangians in a Calabi–Yau manifold. While the existence of continuous (viscosity) solutions is understood, higher regularity of solutions remains a major problem. This talk will discuss some recent progress on the DSL based on joint work with A. Kapota, J.P. Solomon and Y. Rubinstein.