Conway's surreal numbers are a class "No" of inductive objects, originally thought as moves in a game, possessing a natural structure of ordered field and an exponential function which make it a monster model of the theory of (R,exp). It has been conjectured several times that No has also a derivation mimicking the differential structure of Hardy fields, and that No could be described, in some appropriate sense, as a field of transseries. I will discuss the interplay between surreal numbers, derivations of Hardy fields and transseries, and the most recent results regarding the above conjectures. This is joint work with A. Berarducci.