The Atiyah-Singer index theorem expresses the index of an elliptic operator on a closed manifold as the integral of a characteristic class. When the manifold has boundary (and the operator is of first order), there is an additional term in the index formula, the famous Atiyah-Patodi-Singer eta invariant. The APS theorem can be interpreted as an index formula for first-order elliptic operators on a non-compact manifold with a cylindrical end, that is, an end that is a product of a compact manifold with [0,infinity). In a joint work with Tom Mrowka and Nikolai Saveliev, we give an index formula for manifolds that are periodic at infinity. The formula includes a new end-periodic eta invariant. Our index theorem was motivated by considerations of Seiberg-Witten theory on certain non-simply-connected 4-manifolds; I hope to briefly explain this connection.