The Dirichlet problem for the minimal surface equation is known to be solvable for arbitrary continuous boundary values if (and only if) the domain is convex. For the nonconvex problem, di Giorgi discovered a notion that allows generalized solutions for arbitrary boundary values, though the boundary values will not, in general, be achieved in the classical sense. I will survey various questions surrounding the boundary behavior of di Giorgi's generalized solutions and give an example which answers a particular one of those questions posed by John Urbas.