The notion of topological field theory has stymied topologists partly because it assigns to spaces quantities that are multiplicative under disjoint union; traditional homological or homotopical constructions are additive. In this talk I will survey how the use of an old "multiplicative" object in topology ("the spectrum of
units" in the class of vector spaces) leads to a successful formulation of a simple (but non-trivial) 2-dimensional field theory (the "Verlinde ring" and its deformations) and to new topological results about the moduli space of vector bundles on a Riemann surface. This is based on joint work with Freed-Hopkins and with Woodward.