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.

40 pages, v2: journal version