We study the derived categories of coherent sheaves on GushelβMukai varieties. In the derived category of such a variety, we isolate a special semiorthogonal component, which is a K3 or Enriques category according to whether the dimension of the variety is even or odd. We analyze the basic properties of this category using Hochschild homology, Hochschild cohomology, and the Grothendieck group. We study the K3 category of a GushelβMukai fourfold in more detail. Namely, we show this category is equivalent to the derived category of a K3 surface for a certain codimensionΒ 1 family of rational GushelβMukai fourfolds, and to the K3 category of a birational cubic fourfold for a certain codimension 3 family. The first of these results verifies a special case of a duality conjecture which we formulate. We discuss our results in the context of the rationality problem for GushelβMukai varieties, which was one of the main motivations for thisΒ work.