The Batalin-Vilkovisky (BV) formalism provides us with concrete mathematical tools that we can use to describe classical field theories on a manifold, and to make sense of what it means to quantize such a classical theory, all using techniques from homological algebra.  These tools can be applied broadly, but in this talk I'll focus on the special class of "topological" field theories: those that do not depend on additional structure on the base manifold, such as a choice of metric.  In particular, I will discuss an approach to the following general problem.  Given a topological field theory that makes sense on any oriented n-manifold, if we can quantize the theory on R^n, when is it possible to extend to a quantization defined more generally?  In general there is an obstruction to performing this extension (the framing anomaly), and we can characterize this obstruction concretely for a broad class of topological theories (those of "AKSZ type").  This is based on joint work with Owen Gwilliam and Brian Williams.