The notion of a weakly scattered theory T is defined. T need not be scattered. For each $\cal A$ a model of T, let sr($\cal A$) be the Scott rank of $\cal A$. Assume sr($\cal A$) ≤ ω\sp A \sb 1 for all $\cal A$ a model of T. Let σ\sp T \sb 2 be the least Σ₂ admissible ordinal relative to T. If T admits effective k-splitting as defined in this paper, then $∃θ < σ\sp T \sb 2 such that sr($\cal A$) < θ for all $\cal A$ a model of T.

We define some variations of the Scott rank for countable models and obtain some inequalities involving the ranks. For mono-unary algebras we prove that the game rank of any subtree does not exceed the game rank of the whole model. However, similar questions about linear orders remain unresolved.

If some infinitary sentence provides a counterexample to Vaught's Conjecture, then there is an infinitary sentence which also provides a counterexample but has no model of cardinality bigger than ℵ₁.

We offer a topological treatment of scattered theories intended to help to explain the parallelism between, on the one hand, the theorems provable using Descriptive Set Theory by analysis of the space of countable models and, on the other, those provable by studying a tree of theories in a hierarchy of fragments of infinintary logic. We state some theorems which are, we hope, a step on the road to fully understanding counterexamples to Vaught's Conjecture. This framework is in the early stages of development, and one area for future exploration is the possibility of extending it to a setting in which the spaces of types of a theory are uncountable.

We give a model theoretic proof, replacing admissible set theory by the Lopez-Escobar theorem, of Makkai's theorem: Every counterexample to Vaught's Conjecture has an uncountable model which realizes only countably many ℒ$_{ω₁,ω}$-types. The following result is new. Theorem: If a first-order theory is a counterexample to the Vaught Conjecture then it has 2\sp ℵ₁ models of cardinality ℵ₁.

During the Notre Dame workshop on Vaught's Conjecture, Hjorth and Kechris asked which Borel equivalence relations can arise as the isomorphism relation for countable models of a first-order theory. In particular, they asked if the isomorphism relation can be essentially countable but not tame. We show this is not possible if the theory has uncountably many types.

We outline the Hrushovsk-Sokolović proof of Vaught's Conjecture for differentially closed fields, focusing on the use of dimensions to code graphs.

Assume G is a superstable locally modular group. We describe for any countable model M of Th(G) the quotient group G(M) / Gm(M). Here Gm is the modular part of G. Also, under some additional assumptions we describe G(M) / Gm(M) relative to G⁻(M). We prove Vaught's Conjecture for Th(G) relative to Gm and a finite set provided that ℳ(G) = 1 and the ring of pseudoendomorphisms of G is finite.

We work in the context of ω-stable theories. We obtain a natural, algebraic equivalent of ENI-NDOP and discuss recent joint proofs with Shelah that if an ω-stable theory has either ENI-DOP or is ENI-NDOP and is ENI-deep, then the set of models of T with universe ω is Borel complete.

In the last five years there have been a number of results about the computable content of the prime, saturated, or homogeneous models of a complete decidable (CD) theory T in the spirit of Vaught's "Denumerable models of complete theories" combined with computability methods for (Turing) degrees d ≤ 0′. First we recast older results by Goncharov, Peretyat'kin, and Millar in a more modern framework which we then apply. Then we survey recent results by Lange, "The degree spectra of homogeneous models," which generalize the older results and which include positive results on when a certain homogeneous model $\cal A$ of T has an isomorphic copy of a given Turing degree. We then survey Lange's "A characterization of the 0-basis homogeneous bounding degrees" for negative results about when $\cal A$ does not have such copies, generalizing negative results by Goncharov, Peretyat'kin, and Millar. Finally, we explain recent results by Csima, Harizanov, Hirschfeldt, and Soare, "Bounding homogeneous models," about degrees d that are homogeneous bounding and explain their relation to the PA degrees (the degrees of complete extensions of Peano Arithmetic).