Is $\Bbb R$ definable in $(\Bbb C,0,1,+,*,\exp)$?
This is a major open problem in model theory. Recent attempts to study ${\mathbb C}_{\exp}$ have been centred around Zilber's pseudo-exponentiation, which is a nice structure and module some (very serious) algebraic conjectures coincides with ${\mathbb C}_{\exp}$. Zilber's pseudo-exponential field is quasiminimal (i.e. every definable set is countable or co-countable) and hence $\mathbb R$ is not definable there. Have a look at Marker's paper for an introduction.