Special Lagrangian n-folds are high-dimensional volume minimizing submanifolds discovered around 1982 by Harvey and Lawson in their pioneering work on calibrations. More recently special Lagrangians n- folds have played an important role in developments at the interface of geometry and string theory related to Mirror Symmetry. Singular special Lagrangian n-folds play a key role in these development but are still relatively poorly understood. This talk will give an introduction to special Lagrangian geometry and singular special Lagrangian n-folds and will then focus on joint work with Nicos Kapouleas that uses geometric PDE gluing methods to construct a plethora of new singular special Lagrangians (cones) in dimensions 3 and higher.