A remarkable result of Nevanlinna's asserts that a meromorphic function on the complex plane is uniquely determined by its preimages at any five prescribed points. I will present related results in which one know the preimages of some finite sets, rather than the preimages of points. At the same time, I will determine all polynomials f(x) and g(x) with algebraic coefficients for which the equation f(x)=g(y) has infinitely many solutions in some number field. The proofs use various novel Galois-theoretic arguments, including new types of applications of the classification of finite simple groups.