In this paper, we study the complexity of sets formed by boolean operations (union, intersection, and complement) on NP sets. These are the sets accepted by trees of hardware with NP predicates as leaves, and together these form the boolean hierarchy.We present many results about the structure of the boolean hierarchy: separation and immunity results, natural complete languages, and structural asymmetries between complementary classes.We show that in some relativized worlds the boolean hierarchy is infinite, and that for every k there is a relativized world in which the boolean hierarchy extends exactly k levels. We prove natural languages, variations of VERTEX COVER, complete for the various levels of the boolean hierarchy. We show the following structural asymmetry: though no set in the boolean hierarchy is ${\text{D}}^{\text{P}} $-immune, there is a relativized world in which the boolean hierarchy contains ${\text{coD}}^{\text{P}} $-immune sets.Thus, this paper explores the structural properties of the...