I will discuss some aspects of a singular version of the Donaldson-Uhlenbeck-Yau theorem for bundles and sheaves over normal complex varieties satisfying some conditions. Several applications follow, such as a characterization of the case of equality in the Bogomolov-Gieseker theorem. Such singular metrics also arise naturally under certain types of degenerations, and I will make some comments on the relationship between this result and the Mehta-Ramanathan restriction theorem.