Hardy fields and transseries are two among several approaches for enriching the real continuum by infinitesimal and infinite quantities, with applications to dynamical systems and model theory. Both come with natural notions of ordering and differentiation. In joint work with van den Dries and van der Hoeven we developed a framework of ?asymptotic differential algebra? to unify these approaches. In these lectures I will introduce the main objects of interest, explain the basic algebraic framework, sketch the important results obtained so far, and time permitting, outline current challenges and further possibilities.