Abstract. I'm interested in the following problem: let f and g be two continuous strictly increasing and unbounded functions on (0,\infty). Let also f^{-1} and g^{-1} be the corresponding inverse functions. Is it true that f^{-1} and g^{-1} are asymptotically equivalent at infinity if so are f and g? The answer is negative. In this talk I'll review some aspects of Karamata's theory of regularly varying function and discuss its extensions and generalizations. I'll also show in which classes the problem on asymptotic equivalence has a positive solution. Other applications of regularly varying functions will be discussed too.