Value sets of binary forms

Given a binary form F with integer coefficients, we define its value set $Val(F)$ to be the set $F(x, y)$ for integers $x, y$. For two binary forms $F$ and $G$, what can we say if $Val(F) = Val(G)$? The goal of this talk is to show how one may give a complete answer to this question. This is joint work with Etienne Fouvry.