  We give an overview of recent progress around a problem introduced by Elekes
and R\'onyai. The prototype problem is to show that a polynomial $f\in
\mathbb{R}[x,y]$ has a large image on a Cartesian product $A\times B\subset
\mathbb{R}^2$, unless $f$ has a group-related special form. We discuss a number
of variants and generalizations. This includes the Elekes-Szab\'o problem,
which generalizes the Elekes-R\'onyai problem to a question about an upper
bound on the intersection of an algebraic surface with a Cartesian product, and
curve variants, where we ask the same questions for Cartesian products of
finite subsets of algebraic curves. These problems lie at the crossroads of
combinatorics, algebra, and geometry: They ask combinatorial questions about
algebraic objects, whose answers turn out to have applications to geometric
questions involving basic objects like distances, lines, and circles, as well
as to sum-product-type questions from additive combinatorics. As part of a
recent surge of algebraic techniques in combinatorial geometry, a number of
quantitative and qualitative steps have been made within this framework.
Nevertheless, many tantalizing open questions remain.
