Topological field theory provides a very general viewpoint on the notions of dimension, character, trace and charge. I'll explain (based on joint work with David Nadler) how to understand results such as Riemann-Roch theorems, the Atiyah-Bott fixed point theorem, Lefschetz trace formula, character formulas of Frobenius, Weyl, and Harish-Chandra, and the Fundamental Lemma from this perspective.