In this talk I will make a survey, with historical motivations, about the applications of divergent series in order to treat different problems related with differential equations. We will introduce the theory of summability, developed by Malgrange and Ramis among others, and present some recent progresses in this field, related with the treatment of singularly perturbed differential equations. More precisely, the theory of monomial asymptotic expansions will be presented with some applications and possible future developments.