Wilkie's work on the real exponential field showed that there was there was an underlying simplicity to definable sets behaving much like the semialgebaric sets in real algebraic geometry.  By contrast, the integers are definable in the complex exponential field, so  all of the Gödel phenomena are present in this setting.  Nevertheless, there is a hope that the definable sets are somewhat well behaved.  Zilber has proposed an intriguing model theoretic approach which I will describe in this talk.