This chapter is mainly based on the talk presented at the meeting The Philosophy and Physics of Noether’s Theorems that took place 5-6 October 2018, but it also contains some original results that were inspired by discussions with mathematicians, physicists, and philosophers about the problem of understanding the intrinsic meaning of gauge invariance. Following the principles of locality, deformation, and homology, one naturally ends up using the Batalin-Vilkovisky (BV) formalism in quantizing gauge theories. Beginning with the gentle introduction into the BV framework, the chapter proceeds to some new results and more speculative deliberations. In the classical theory, a new perspective is presented on the classical BV operator, using the notion of Møller maps. In the quantum theory, some loose ideas are presented on the formulation of anomalous master Ward identity in the framework proposed recently by Buchholz and Fredenhagen, based on local S-matrices.