  In this paper, we first prove a complete version of the
Donaldson-Uhlenbeck-Yau theorem over normal varieties, including normal Kaehler
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 Kaehler 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, as another application, we give a new
criteria for when a normal Kaehler variety with trivial first Chern class is a
finite quotient of torus.
