Hilbert's tenth problem for finitely generated rings

This expository article covers the recent developments surrounding Hilbert's tenth problem for finitely generated rings. We start by recounting the history of Hilbert's tenth problem over the integers, which was resolved negatively by Matiyasevich--Robinson--Davis--Putnam in 1970. In order to pass from $\mathbb{Z}$ to the finitely generated setting, we explain a criterion of Poonen that connects this to a problem in the theory of elliptic curves. Finally, we outline the main ideas behind the recent resolution of this elliptic curve problem by the authors.