The Pila--Wilkie Theorem provides a subpolynomial bound on the number of rational points of bounded height lying on the `transcendental parts' of sets $X$ definable in o-minimal expansions of the real field (that is, for any $\epsilon>0$ and height $H\geq 1$, the bound is of the form $\ll_{X, \epsilon} H^\epsilon$).

The pursuit of examples in which one can obtain an effective implied constant in this bound is very active, with a view to applications. I will discuss some recent progress made in this direction in the case of surfaces implicitly defined from restricted Pfaffian functions. These have a natural notion of complexity, and we show that, in our setting, an effective implied constant exists which is uniform in this complexity. (Joint work with Gareth O. Jones.)