Hilbert's Tenth Problem for function fields of characteristic zero

In this article we outline the methods that are used to prove undecidability of Hilbert's Tenth Problem for function fields of characteristic zero. Following Denef we show how rank one elliptic curves can be used to prove undecidability for rational function fields over formally real fields. We also sketch the undecidability proofs for function fields of varieties over the complex numbers of dimension at least 2.