This is a survey of the authors' results on the properties and applications of some subclasses of (so-called) $O$-regularly varying (ORV) functions. In particular, factorization and uniform convergence theorems for AvakumoviÄ‡--Karamata functions with non-degenerate groups of regular points are presented together with the properties of various other extensions of regularly varying functions. A discussion of equivalent characterizations of such classes of functions is also included as well as that of their (asymptotic) inverse functions. Applications are given concerning the asymptotic behavior of solutions of certain stochastic differential equations.