The Rokhlin lemma is a finite approximation property that underpins a great many constructions in classical ergodic theory, including most spectacularly those at the basis of the Ornstein isomorphism theory for Bernoulli shifts. In the 1970s Ornstein and Weiss showed amenability to be the natural setting for finite approximation in dynamics by establishing a general form of the Rokhlin lemma in this setting, and this led, among other things, to a much broader recasting of the Ornstein isomorphism theory. Over the last couple of decades a growing interest in the interplay between dynamics and the geometric and analytic structure of groups has set the stage for a resurgence of applications of the Ornstein-Weiss Rokhlin lemma, not only in its original measure-theoretic incarnation but also as a versatile tiling principle that has turned out be intimately connected, on the topological side, to the remarkable recent successes in the Elliott classification program for separable nuclear C*-algebras. I will sketch a picture of these various developments at the interface of measure, topology, dynamics, geometric group theory, and operator algebras.