Abstract: The "signature" or eta invariant originally appeared as the boundary correction term of the famous index formula of Atiyah, Patodi, and Singer, but since this seminal paper, the study of the eta invariant has taken off with a life of its own. In this talk, I will give a very gentle introduction to the Atiyah-Patodi-Singer index formula, then I'll focus on the eta invariant and some of its properties, but for the bulk of the talk I'll discuss some remarkable properties of manifolds that have cone points and their eta invariants.