In this paper we prove three theorems about first-order theories that are categorical in a higher power. The first theorem asserts that such a theory either is totally categorical or there exist prime and minimal models over arbitrary base sets. The second theorem shows that such theories have a natural notion of dimension that determines the models of the theory up to isomorphism. From this we conclude that $I(T, \aleph_\alpha) = \aleph_0 +|\alpha|$ where ℵ α = the number of formulas (...) modulo T-equivalence provided that T is not totally categorical. The third theorem gives a new characterization of these theories. (shrink)