Perhaps most notorious for its involvement in the Banach-Tarski paradox, the notion of amenability touches a broad spectrum of disciplines, such as combinatorial and geometric group theory, probability, ergodic theory, and set theory. While typically cast as a property of groups, it turns out that many key aspects of amenability are captured by the orbit equivalence relation of a measurable group action, which in general carries much less information than the group itself. We'll define and discuss these notions along with some of the major theorems and applications, and also collect some of the (many!) open problems in the area.