. Originating in harmonic analysis, interpolation sets were first studied in dynamics by Glasner and Weiss in the 1980s [Israel J. Math. 44 (1983), pp. 345-360]. A set S subset of N is an interpolation set for a class of topological dynamical systems C if any bounded sequence on S can be extended to a sequence that arises from a system in C. In this paper, we provide combinatorial characterizations of interpolation sets for: center dot (totally) minimal systems; center dot topologically (weak) mixing systems; center dot strictly ergodic systems; and center dot zero entropy systems. Additionally, we prove some results on a slightly different notion, called weak interpolation sets, for several classes of systems. We also answer a question of Host, Kra, and Maass [Monatsh. Math. 179 (2016), pp. 57-89] concerning the connection between sets of pointwise recurrence for distal systems and IP-sets.