Petr Naryshkin of the Institute for Advanced Study gives this Joint IAS/PU Groups and Dynamics Seminar talk on extending the Rokhlin Lemma from ergodic theory into topological dynamics. He starts with the classical measurable result, which builds a tower of sets covering almost all of a probability space under an aperiodic transformation, and the Ornstein-Weiss generalization to amenable group actions. He then turns to the topological setting, where an amenable group acts by homeomorphisms on a compact metrizable space, explaining why several competing formulations exist and why genuine obstructions arise from both the group's algebraic structure and the space's topology. The talk covers applications to C*-crossed products, mean dimension theory, and Naryshkin's own work on shift embeddability, closing with conditions (developed partly with Petrakos) under which topological Rokhlin-type properties can be proved. The audience is researchers in ergodic theory and operator algebras.