Filip Najman of the University of Zagreb presents this joint IAS/Princeton University Number Theory Seminar talk on classifying rational points on modular curves, a problem tied to Mazur's Program B for describing the Galois images of elliptic curves over Q. Najman explains why the classification problem is infinite, since infinitely many modular curves carry rational points, some with finitely many and some with infinitely many. Building on Zywina's notion of agreeable subgroups, he introduces twist-parametrized and twist-isolated points, and shows that under standard conjectures on Galois representations, every non-cuspidal, non-CM rational point on any modular curve is twist-parametrized by points on a specific list of 160 modular curves. The work is joint with Maarten Derickx, Sachi Hashimoto, and Ari Shnidman. Delivered at Simonyi Hall, the talk is aimed at researchers in arithmetic geometry and number theory already familiar with elliptic curves and modular curves.