In this paper, we prove a singular version of the Donaldson-Uhlenbeck-Yau theorem over normal projective varieties and normal complex subvarieties of compact Kahler manifolds that are smooth outside a codimension three analytic subset. As a consequence, we deduce the polystability of (dual) tensor products of stable reflexive sheaves, and we give a new proof of the Bogomolov-Gieseker inequality over such spaces, along with a precise characterization of the case of equality. In addition, we improve several previously known algebro-geometric results on normalized tautological classes. We also study the limiting behavior of semistable bundles over a degenerating family of normal projective varieties. In the case of a family of stable bundles, we explain how the singular Hermitian-Yang-Mills connections obtained here fit into the degeneration picture. These can also be characterized from the algebro-geometric perspective. As an application, we apply the results to the degeneration of stable bundles through the deformation to projective cones, and we explain how our results are related to the Mehta-Ramanathan restriction theorem.
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