I will talk about a recent result showing that some well-studied polynomial-based error-correcting codes (Folded Reed-Solomon Codes and Multiplicity Codes) are "list-decodable upto capacity with constant list-size". At its core, this is a statement about questions of the form: "Given some points in the plane, how many low degree univariate polynomials are such that their graphs pass through 10% of these points"? This leads to list-decodable and locally list-decodable error-correcting codes with the best known parameters. Based on joint work with Noga Ron-Zewi, Shubhangi Saraf and Mary Wootters.