BF theory does not quite fit into (strict) Atiyah’s axioms. The space of states it assigns to a boundary is typically infinite-dimensional (which implies that the partition function of $S^1 \times X$ is infinite). This can be seen (a) as a consequence of noncompactness of the phase space of the theory or (b) as a manifestation of the problem of zero-modes. The BV-BFV formalism is an approach to gauge theories (in particular, topological ones) combining the Atiyah-Segal functorial picture with the idea of Wilson’s effective action. In this talk I will sketch the construction of BF theory in the BV-BFV language and will explain how it assigns meaningful partition functions (satisfying an appropriate gluing property) to all cobordisms.