Counting algebraic points on definable sets

Pila and Wilkie's influential theorem, which bounds the density of rational and algebraic points lying on the transcendental parts of sets definable in o-minimal expansions of the real field, has already had far-reaching consequences for diophantine geometry. While their result gives the best bound possible for o-minimal structures in general, Wilkie has conjectured an improvement for sets definable in the real exponential field, namely that the bound could be improved to one involving some power of the logarithm of the height. This conjecture has already been established for curves and for certain surfaces. While the full conjecture seems distant, there are already some interesting applications of known results in this direction. We will outline briefly what has been established and discuss some of the consequences.