When is a Schubert variety Gorenstein?

A (normal) variety is Gorenstein if it is Cohen-Macaulay and its canonical sheaf is a line bundle. This property is a weakening of smoothness which implies that a variety behaves like a smooth one for various algebro-geometric purposes. We give a combinatorial characterization of which Schubert varieties are Gorenstein, using in part a new generalization of pattern avoidance. We also give an explicit description as a line bundle of the canonical sheaf of a Gorenstein Schubert variety.