<p>In this paper, we first prove a complete version of the Donaldson–Uhlenbeck–Yau theorem over normal varieties, including normal Kähler varieties and projective normal varieties with multiple polarizations. In particular, this gives the polystability of reflexive sheaves under symmetric and exterior powers and tensor products. As a consequence of the singular Donaldson–Uhlenbeck–Yau theorem, the complete Hitchin–Kobayashi correspondence over normal varieties smooth in codimension two is built by showing that an admissible Hermitian–Yang–Mills connection defines a polystable reflexive sheaf. Furthermore, it is shown that the Hermitian–Yang–Mills connection gives a lower bound for the discriminants of any Kähler resolutions, which gives a Bogomolov–Gieseker inequality over normal varieties and a characterization of the equality using projectively flat connections. We discuss typical cases including normal surfaces and varieties smooth in codimension two where we could simplify the Bogomolov–Gieseker inequality and endow it with topological meanings. We also prove the Bogomolov–Gieseker inequality for semistable reflexive sheaves and characterize the class of semistable sheaves that satisfy the Bogomolov–Gieseker equality. Finally, we give a new criteria for when a normal Kähler variety with trivial first Chern class is a finite quotient of torus.</p>