I will discuss results connected to the asymptotic density of rational
points on transcendental varieties, starting with the theorem of
Pila-Wilkie on general o-minimal structures and some of its diophantine
applications. I will then discuss the Wilkie conjecture asserting a
significant sharpening of the P-W theorem for the structure R_\exp, and a
recent proof (joint with Novikov) of this conjecture for sets definable
using only restricted exponentiation (and restricted sine). I will briefly
discuss the proof, and in particular the significant role played by the
algebraic differential equations for exp and sin. Finally, I will discuss
some related questions of diophantine significance for structures generated
by elliptic, modular and other special functions, and what role the
differential-algebraic structure of these functions may play in resolving
these questions.