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The BV-BFV formalism, developed by Cattaneo, Mnev and Reshetikhin (CMR), is a powerful tool to understand quantisation of field theories on manifolds with boundaries. It combines the known BV (Batalin, Vilkovisky) framework for gauge theories on closed manifolds, which generalises the older BRST (Becchi, Rouet, Stora and Tyutin) machinery, with its Hamiltonian counterpart on boundaries (BFV, with Fradkin), in the direction of an axiomatic understanding of quantum field theory.
One of the features of the formalism is how it handles both symmetry and boundary data of a given classical field theory to set up a clean quantisation scheme. While doing so, it encodes all of the relevant physical information about the theory and gives a clear handle on the classical theory, possibly beyond the standard Hamiltonian approach to field theory.

The scheme proposed by CMR is very promising, as the examples of Chern Simons, BF and Poisson Sigma model have shown, and testing it on a theory like General Relativity is one interesting challenge.
In this talk I will present the basics of the BV-BFV formalism, using simple one-dimensional theories as running examples. This will help us to approach General Relativity, to describe our current understanding of the problem, and the new, sometimes unexpected results that we get from this one-century-old problem.