Delve into the intricacies of the BV-BFV program in this comprehensive lecture, the third installment of a series presented by Pavel Mnev at the Galileo Galilei Institute. Explore advanced concepts and applications of the Batalin-Vilkovisky and Batalin-Fradkin-Vilkovisky formalisms in quantum field theory and gauge systems. Gain insights into the mathematical structures and physical interpretations underlying this powerful framework for handling gauge theories and constrained systems. Enhance your understanding of modern approaches to quantum field theory and gauge theory quantization through this in-depth, nearly two-hour presentation.