Let F(x,y) be an irreducible polynomial over Q, and let C be the plane curve cut out by F(x,y) = 0. We say that a number field K/Q is generated by F if there exists an algebraic point P on C such that K = Q(P). Recently, Mazur and Rudin suggested the idea of studying the geometry of C by instead investigating the set of all number fields generated by C. This relationship can also be examined in the opposite direction, by fixing a curve or a family of curves and asking what can be said about the fields generated by that curve or family. In particular, when C is an elliptic curve, Lemke Oliver and Thorne have given lower bounds for the proportion of degree n fields generated by C whose Galois closure has Galois group isomorphic to S_n. This has been extended to hyperelliptic curves by Keyes. We discuss the general approach laid out by Keyes, as well as joint work in progress with Bell, Lemke Oliver, Serrano López, and Wong to generalize this approach to arbitrary plane curves.