  This is a survey of our program of perturbative quantization of gauge
theories on manifolds with boundary compatible with cutting/pasting and with
gauge symmetry treated by means of a cohomological resolution
(Batalin-Vilkovisky) formalism. We also give two explicit quantum examples --
abelian BF theory and the Poisson sigma model. This exposition is based on a
talk by P.M. at the ICMP 2015 in Santiago de Chile.
