Abstract: The notion of hyperfiniteness is a central one in both logic and dynamics, and forms a common point where the two subjects meet. Many important questions (e.g., the hyperfiniteness conjecture) about this notion remain open, however. We will introduce the notion, discuss early results, and present some recent work of Gao and Jackson on the hyperfiniteness of abelian group actions (later generalized by Seward and Schneider).