Abstract: After a quick review of the main results on moduli spaces of semistable sheaves on K3 surfaces we will recall a fundamental theorem of Verbitsky. The latter result opens the way to an extension to higher dimensional IHSM's of one key result valid for  K3 surfaces, provided we consider vector bundles with certain characteristic classes. Lastly we will illustrate this principle by stating a recent result of ours on moduli spaces of stable vector bundles on IHSM's of Type K3^[2] (deformations of the Hilbert square of a K3 surface). If time permits I will sketch the proof.