This is an expository paper on a Theorem of the Complement, due to Wilkie, and its generalisations. Wilkie (Sel Math (NS) 5:397–421, 1999) gave necessary and sufficient conditions for an expansion of the real field by C-infinity functions to be o-minimal. Karpinski and Macintyre (Sel Math (NS) 5:507–516, 1999) weakened the original smoothness hypotheses of Wilkie’s theorem. Here we exhibit the proof of a generalised Wilkie’s result, where we further weaken the smoothness assumptions and show that the proof can be carried out not only over the real numbers but more generally in a non-Archimedean context, i.e. for definably complete Baire structures, which we introduced in 2008 and which form an axiomatizable class. Furthermore we give necessary and sufficient conditions for a definably complete Baire expansion of an o-minimal structure by C-infinity functions to be o-minimal.