Abstract
Let $T$ be an o-minimal theory extending the theory of the real numbers. To any model $R$ of $T$ one can attach a subring $V$ and then study the theory of the pair $(R,V)$. Under some assumptions on $V$ ($T$-convexity), this theory turns out to have very interesting properties ranging from quantifier elimination and existence of definable Skolem functions to weakly o-minimality. I will present some of the results of L. van den Dries and A. H. Lewenberg on this theory.