The principle of gauge invariance and the quantum paradigm are two cornerstones of modern fundamental physics and geometry. The intricate relation of the two underlies various challenges in the physics of fundamental interactions and modern developments in mathematics ranging from quantum gravity and topological effects in quantum physics to homological algebra, super geometry and derived geometry. The by now classical works of Igor Batalin with E. Fradkin and G. Vilkovisky targeted the very compatibility of the gauge invariance and quantization, resulting in a very general mathematical formalism which by now is considered as a far going generalisation of the usual Lagrangian/Hamiltonian framework at both classical and quantum level. The applications of the approach of Batalin, Fradkin and Vilkovisky expanded far beyond the original scope of gauge theories quantization and the approach itself is more and more considered as a proper language to define quantum gauge field theories, to look for new theories of fundamental interactions and to study geometrical structures.

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