Abstract
The results outlined in this talk are part of a wider, flourishing interaction between diophantine geometry and model theory. The central aim is to bound the density of rational and algebraic points lying on certain `transcendental' subsets of the reals, with a view to diophantine applications. Following influential work by Pila and Wilkie in this area (which has had stunning number-theoretic applications e.g. to the Manin-Mumford and André-Oort Conjectures), our focus is on sets which are first-order definable in various `o-minimal' expansions of the real field. We shall survey background and some results in this area, in particular concerning possible improvements to the bound given by Pila and Wilkie. These include instances of a conjecture of Wilkie, which proposes an improvement for the real exponential field, and some recent progress made towards finding an effective version of the Pila-Wilkie Theorem.