Singularities of special Lagrangian submanifolds

We survey what is known about singularities of special Lagrangian submanifolds (SL m-folds) in (almost) Calabi-Yau manifolds. The bulk of the paper summarizes the author’s five papers math.DG/0211294, math.DG/0211295, math.DG/0302355, math.DG/0302356, math.DG/0303272 on SL m-folds X with isolated conical singularities. That is, near each singular point x, X is modelled on an SL cone C in C^m with isolated singularity at 0. We also discuss directions for future research, and give a list of open problems.