We characterize those Gorenstein algebraic varieties which are seminormal (in the sense of [11]), by describing their singularities in codimension 1. In particular a plane curve is seminormal if, and only if, it has at most ordinary double points (as proved by P. Salmon in [10]); and a surface in 3-space is seminormal if, and only if, it has at most "biplanar" double curves. It follows that a surface with "ordinary singularities" only is seminormal, as proved by E. Bombieri in [5].