The correspondence between polystable reflexive sheaves on compact Kaehler manifolds and the existence of suitably singular Hermitian-Einstein metrics can be extended to normal projective varieties that are smooth in codimension two. A particular application is a characterization of those sheaves which saturate the Bogomolov-Gieseker inequality. This talk will present some of the key details of this result, which is joint work with Xuemiao Chen.