The results surveyed in this talk are part of a wider, flourishing interaction between diophantine geometry and o-minimality. The central problem is to bound the density of rational and algebraic points on certain subsets of the reals. Beginning with work by Bombieri and Pila in the 1980s, a key breakthrough came from the application of o-minimality to the problem by Pila and Wilkie in 2006. This, in particular thanks to a strategy utilizing their result developed by Pila and Zannier, has found many significant applications to important open problems in number theory.
Our current focus is on a conjecture of Wilkie which would significantly improve the work of Pila and Wilkie in certain (o-minimal) settings. We will outline the context and the key results, including the progress made so far towards Wilkie's conjecture; of particular interest will be the role played by certain ''smooth parameterizations'' - a geometrical property which also arises in smooth dynamics and computational geometry.