In this paper we introduce three new classes of functions under names translational slowly varying, translational regularly varying and translational $\mathcal{O}$-regularly varying functions. All classes have important applications in the study of asymptotic processes. In this sense, Uniform Convergence Theorem, Characterization Theorem and Representation Theorem are the main results of this paper for all cite classes of functions. This results are closely connected with the Karamata's theory of regularly varying functions. Also, in this paper we introduce three classes of sequences under names translational slowly varying, translational regularly varying and translational $\mathcal{O}$-regularly varying sequences. All three classes have important applications in the study of asymptotic processes. The results are of relevance in connection with limit statements in various branches of probability theory and ergodic theory.