In 1961 Robert Vaught published a ground breaking paper on countable model theory entitled “Denumerable Models of Complete Theories”. In this paper he set out many of the fundamental results concerning countable model theory. At the end of this paper he made a conjecture that every countable first order theory in every model of set theory had either countably or continuum many countable models. This conjecture is what is now called “Vaught’s Conjecture” and is one of the oldest open problems in model theory. Over the years there have been several variants of Vaught’s conjecture which have been shown to be equivalent to it or to be implied by it. In the first part of this series of talks we will show that Vaught’s conjecture is equivalent to the following statement: “If a countable first order theory T does not have a perfect kernel of countable models, then every model of T is isomorphic to a model in L[T]” In other words, if there isn’t a natural relationship between the countable models of T and the real numbers (as in for example the theory (Z, Succ, 0, P) where P is some subset of Z) then the countable models of T are independent of the set theory you are working in (for example the theory of algebraically closed fields of characteristic 0). We call a sentence of L_{\omega_1, \omega} which does not have a perfect kernel of models “scattered” So Vaught’s conjecture, whether true or false, has important implications for the connections between set theory and countable model theory. We will show that these two statements of Vaught’s conjecture are equivalent by constructing what is known as the Vaught tree for any sentence of L_{\omega_1, \omega} and discussing properties of this tree. The second part of this series of talks will then focus on presenting the ideas behind the central result of my dissertation, that for every countable ordinal \alpha there is a scattered sentence of L_{\omega_1,\omega} of quantifier rank \omega which has a Vaught tree of at least height \alpha.