The Verlinde formula is a celebrated explicit computation of the dimension of the space of sections of certain positive line bundles over the moduli space of semistable vector bundles over an algebraic curve. I will describe a recent generalization of this formula in which the moduli of vector bundles is replaced by the moduli of semistable Higgs bundles, a moduli space of great interest in geometric representation theory. A key part of the proof is a new ``virtual non-abelian localization formula" in K-theory, which has broader applications in enumerative geometry. The localization formula is an application of the nascent theory of Theta-stratifications, and it serves as a new source of applications of derived algebraic geometry to more classical questions.