Many of the most naturally occurring spaces have a surprisingly simple cell structure, having cells in only even dimensions. Complex Grassmanians, Thom spaces, and loops on a Lie group all have this property. Dually, one can consider generalized cohomology theories like complex K-theory with only even homotopy groups. Evaluating these cohomology theories on spaces with only even cells is extremely simple, and this interplay drives much of modern algebraic topology. I'll describe some classical work in this area before exploring some of the more recent, equivariant generalizations.