Let G be a finite group and let F be a family of subgroups of G. We introduce a class of G-equivariant spectra that we call F-nilpotent. This definition fits into the general theory of torsion, complete, and nilpotent objects in a symmetric monoidal stable ∞-category, with which we begin. We then develop some of the basic properties of F-nilpotent G-spectra, which are explored further in the sequel to this paper. In the rest of the paper, we prove several general structure theorems for ∞-categories of module spectra over objects such as equivariant real and complex K-theory and Borel-equivariant MU. Using these structure theorems and a technique with the flag variety dating back to Quillen, we then show that large classes of equivariant cohomology theories for which a type of complex-orientability holds are nilpotent for the family of abelian subgroups. In particular, we prove that equivariant real and complex K-theory, as well as the Borel-equivariant versions of complex-oriented theories, have this property.