Introduction. A theory, 1, (formalized in the first order predicate calculus) is categorical in power K if it has exactly one type of models of power K. This notion was introduced by Los [ 9] and Vaught [ 16] in 1954. At that time they pointed out that a theory (e.g., the theory of dense linearly ordered sets without end points) may be categorical in power N0 and fail to be categorical in any higher power. Conversely, a theory may be categorical in every uncountable power and fail to be categorical in power N0 (e.g., the theory of algebraically closed fields of characteristic 0). Los' then raised the following question. Is a theory categorical in one uncountable power necessarily categorical in every uncountable power? The principal result of this paper is an affirmative answer to that question. We actually prove a stronger result, namely: If a theory is categorical in some uncountable power then every uncountable model of that theory is saturated. (Terminology used in the Introduction will be defined in the body of the paper; roughly speaking, a model is saturated, or universalhomogeneous, if it contains an element of every possible elementary type relative to its subsystems of strictly smaller power.) It is known(2) that a theory can have (up to isomorphism) at most one saturated model in each power. It is interesting to note that our results depend essentially on an analogue of the usual analysis of topological spaces in terms of their derived spaces and the Cantor-Bendixson theorem. The paper is divided into five sections. In ?1 terminology and some meta-mathematical results are summarized. In particular, for each theory, 1, there is described a theory, *, which has essentially the same models as z but is neater to work with. In ?2 is defined a topological space, S(A), corresponding to each subsystem, A, of a model of a theory, 1; the points of S(A) being the isomorphism types of elements with respect to A. With each monomorphism (= isomorphic imbedding), f: A -+ B, is associated a dual continuous map, f*: S(B) -+ S(A). Then there is defined for each S(A) a decreasing sequence ISa(A) I of subspaces which is analogous to (but different from)