Overlap functions and grouping functions are generalizations of continuous t-norms and t-conorms, respectively, when associativity is dropped. These functions can be applied in decision making, classification or image processing. In this paper, the authors introduce interval overlap functions and interval grouping functions and show their properties, including construction methods. The authors study in detail interval additive generators of interval overlap functions and interval grouping functions and prove that the best interval representation of the additive generator pair of an additively generated overlap function is an interval additive generator pair of the best interval representation of this overlap function (and analogously for interval grouping functions). The results in this paper allow to construct appropriate interval overlap or grouping functions for individual applications more easily.