  Basic properties of gauge theories in the framework of Faddeev-Popov (FP)
method, Batalin-Vilkovisky (BV) formalism, functional renormalization group
(FRG) approach are considered. The FP and BV quantizations are characterized by
the Becchi-Rouet-Stora-Tyutin (BRST) symmetry while the BRST symmetry is broken
in the FRG approach. It is shown that the FP method, the BV formalism and the
FRG approach can be provided with the Slavnov-Taylor identity, the Ward
identity and the modified Slavnov-Taylor identity, respectively. It is proven
that using the background field method the background gauge invariance of
effective action within the FP and FRG quantization procedures can be achieved
in nonlinear gauges. The gauge-dependence problem within the FP, BV and FRG
quantizations is studied. Arguments allowing us to state the existence of
principal problems of the FRG in the case of gauge theories are given.
