  In this paper we extend the classical BV framework to gauge theories on
spacetime manifolds with boundary. In particular, we connect the BV
construction in the bulk with the BFV construction on the boundary and we
develop its extension to strata of higher codimension in the case of manifolds
with corners. We present several examples including electrodynamics, Yang-Mills
theory and topological field theories coming from the AKSZ construction, in
particular, the Chern-Simons theory, the $BF$ theory, and the Poisson sigma
model. This paper is the first step towards developing the perturbative
quantization of such theories on manifolds with boundary in a way consistent
with gluing.
