Abstract: Given an arbitrary set of real numbers A and a two-variate polynomial f with real coefficients, a remarkable theorem of Elekes and R\'onyai from 2000 states that the size |f(A,A)| of the image of f on the cartesian product A x A grows asymptotically faster than |A|, unless f exhibits additive or multiplicative structure. Finding the best quantitative bounds for this intriguing phenomenon (and for variants of it) has generated a lot of interest over the years due to its intimate connection with the sum-product problem from additive combinatorics. In this talk, we will quickly review some of the results in this area, and then discuss some new bounds for |f(A,A)| when the set A has few sums or few products. If time permits, will also discuss some new results over finite fields.