Fixed point theory was a motivation for some of the major results of 20th century mathematics. For example, étale cohomology and the Atiyah-Singer index theorem both had their genesis in the Lefschetz fixed point theorem. In this talk, I'll describe how an important K-theoretic invariant, the cyclotomic trace, is surprisingly also related to fixed point theory. This description requires worthwhile journeys through algebraic K-theory, the stable homotopy category, and duality theory.