In their 2006 seminal paper, Pila and Wilkie proved that if X is a subset of R^n definable in an o-minimal structure, and it has "many" rational points then X contains a piece if a real algebraic curve.
Following a strategy of Pila and Zannier, that theorem became an important component in the proofs of various results of similar flavor in arithmetic geometry. Examples are the Manin-Mumford conjecture (originally proved by Raynaud), and the Andre-Oort conjecture (proved by Pila).<br>

In this talk I will survey the basic ingredients of these results, and present the general strategy of Pila and Zannier.