Abstract. We prove that from categoricity in λ + we can get categoricity in all cardinals ≥ λ + in a χ-tame abstract elementary classes which has arbitrarily large models and satisfies the amalgamation and joint embedding properties, provided λ > LS(K) and λ ≥ χ.For the missing case when λ = LS(K), we prove that K is totally categorical provided that K is categorical in LS(K) and LS(K)+ .

The results in this paper are in a context of abstract elementary classes identified by Shelah and Villaveces in which the amalgamation property is not assumed. The long-term goal is to solve Shelah's Categoricity Conjecture in this context. Here we tackle a problem of Shelah and Villaveces by proving that in their context, the uniqueness of limit models follows from categoricity under the assumption that the subclass of amalgamation bases is closed under unions of bounded, ≺Kincreasing chains.

Abstract. We introduce tame abstract elementary classes as a generalization of all cases of abstract elementary classes that are known to permit development of stability-like theory. In this paper we explore stability results in this new context. We assume that K is a tame abstract elementary class satisfying the amalgamation property with no maximal model. The main results include:where κµ(K) is a relative of κ(T ) from first order logic. Hanf(K) is the Hanf number of the class K. It is known that Hanf(K) ≤ (2 LS(K) ) +The theorem generalizes a result from [Sh3]. It is used to prove both the existence of Morley sequences for non-splitting (improving Claim 4.15 of [Sh 394] and a result from [GrLe1]) and the following initial step towards a stability spectrum theorem for tame classes:Theorem 0.2. If K is Galois-stable in some µ > (2 Hanf (K) ) + , then K is stable in every κ with κ µ = κ. E.g. under GCH we have that K Galois-stable in µ implies that K is Galois-stable in µ +n for all n < ω.

Abstract. We prove:Main Theorem: Let K be an abstract elementary class satisfying the joint embedding and the amalgamation properties. Let µ be a cardinal above the the Löwenheim-Skolem number of the class. Suppose K satisfies the disjoint amalgamation property for limit models of cardinality µ. If K is µ-Galois-stable, has no µ-Vaughtian Pairs, does not have long splitting chains, and satisfies locality of splitting, then any two (µ, σ ℓ )-limits over M for (ℓ ∈ {1, 2}) are isomorphic over M .This

Abstract. This paper is part of a program initiated by Saharon Shelah to extend the model theory of first order logic to the nonelementary setting of abstract elementary classes (AECs). An abstract elementary class is a semantic generalization of the class of models of a complete first order theory with the elementary substructure relation. We examine the symmetry property of splitting (previously isolated by the first author) in AECs with amalgamation that satisfy a local definition of superstability.The key results are a downward transfer of symmetry and a deduction of symmetry from failure of the order property. These results are then used to prove several structural properties in categorical AECs, improving classical results of Shelah who focused on the special case of categoricity in a successor cardinal.We also study the interaction of symmetry with tameness, a locality property for Galois (orbital) types. We show that superstability and tameness together imply symmetry. This sharpens previous work of Boney and the second author.