Perturbative BV theories with Segal-like gluing

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.