An important invariant associated to a first order theory T is its spectrum: the number of models of T of each cardinality. Sixty years ago, Vaught conjectured that the number of countable models of a complete theory T is always either countable or 2^{\aleph_0}. While Vaught’s conjecture remains open, significant partial results have been obtained, e.g. Morley’s proof that any counterexample to Vaught’s conjecture must have cardinality exactly \aleph_1. Morley’s proof uses techniques from descriptive set theory, and easily generalizes to sentences of the infinitary logic L_{\omega_1,\omega}. We will discuss the history of Vaught’s conjecture, including the many closely related conjectures that it has spawned and the current status of the conjecture.