Abstract: O-minimal structures are categories of sets and mapping having nice geometrical properties. To each o-minimal expansion of a real closed field, one can associate the set of germs at infinity of its unary functions, which form a Hardy field. Valuational properties of these Hardy fields give good information about the initial structure. After a lengthy introduction of all the previously named objets and motivated by a conjecture of L. van den Dries and a result of F.-V. and S. Kuhlmann, I will discuss whether an o-minimal expansions of the field of the reals is, in general, fully determined by its associated Hardy field. I will also relate this question to the Hilbert's 13th Problem.