The Event Horizon Telescope image of the supermassive black hole in the galaxy M87 is dominated by a bright, unresolved ring. General relativity predicts that embedded within this image lies a thin “photon ring,” which is itself composed of an infinite sequence of self-similar subrings. Each subring is a lensed image of the main emission, indexed by the number of photon orbits executed around the black hole. I will review recent theoretical advances in our understanding of lensing by Kerr black holes, based on arXiv preprints 1907.04329, 1910.12873, and 1910.12881. In particular, I will describe the critical parameters γ, δ, and τ that respectively control the demagnification, rotation, and time delay of successive lensed images of a source, thereby providing an analytic theory of the photon ring. These observable parameters encode universal effects of general relativity, which are independent of the details of the emitting matter and also produce strong, universal signatures on long interferometric baselines. These signatures offer the possibility of precise measurements of black hole mass and spin, as well as tests of general relativity, using only a sparse interferometric array such as a future extension of the EHT to space.