Tarski's work from the 1930's, on the structure of first-order definitions in the real and complex fields, eventually became a standard tool in semi-algebraic geometry. Tarski (who knew that the complex exponential was "wild" in the sense of interpreting arithmetic) posed the issue of analyzing effectively the real exponential along the lines of his analysis for the real field. This problem took well over 50 years to solve, and general ideas isolated during its solution (the theory of 0-minimality) have proved applicable to deep matters in Lie theory and, more recently, in diophantine geometry. The decisive result was proved by Wilkie in 1991 (depending on Hovanski's work) , and Tarski's original problem on effectivity was solved by Wilkie and Macintyre in 1992, ASSUMING the truth of Schanuel's Conjecture in transcendental number theory .
Now, even Schanuel's Conjecture cannot make the complex exponential decidable, but very deep work of Zilber has led to the unconditional construction of exponential fields ("Zilber fields") which have an amazingly structured model theory, satisfy Schanuel's Conjecture, and share quite a few properties of the complex exponential. One striking feature is that the properties in question are usually proved for the complexes by difficult analysis, and for the Zilber fields by relatively simple algebra. Zilber has conjectures that the complex exponential field is a Zilber field. This cannot be true unless Schanuel's Conjecture is true, but if one assumes that then the conjecture yields many new insights about complex analysis of exponential functions.
It turns out that Zilber's model theory is deeply connected to issues in diophantine geometry studied initially by Bombieri, Masser and Zannier. It turns out too that there are connections to old problems in complex analysis, for example Shapiro's Conjecture on common zeros of exponential functions. I will discuss how this is related to Schanuel's Conjecture.
A big mystery, where the best we know is due to Vincenzo Mantova, is how to detect in Zilber fields something like the Euclidean topology. I will discuss this.