Hochschild homology is a categorification of the trace. We will start by explaining what this means, and how both the trace and Hochschild homology are relevant to topological field theories. We will then explain an algebraic result for Hochschild homology, and how it can sometimes be used to study TQFT invariants of covering spaces. The first half of this talk is classical; the second half is joint work with D. Treumann.