There are several different notions of what it might mean for a space to be “even’’ in algebraic topology, all of which have useful, increasingly algebraic properties. I will begin by describing some classical work due to Wilson, focusing on what sorts of things happen when we have only even cells or homotopy groups, and then I will describe more recent work (all joint with Hopkins), where building on geometric intuition, we construct a version of this that works also when there is an action of a finite group.