Vaught's conjecture on the number of countable models

Let $T$ be an elementary first-order theory, and consider its countable models up to isomorphism. Vaught's conjecture. The number of countable models of $T$ is either countable or continuum. This is a central question in model theory. Morley proved that the number is one of $\a0\aleph_0$, $\a0\aleph_1$, or continuum, so the conjecture rules out the intermediate possibility $\a0\aleph_1$; it is automatically true under the continuum hypothesis and remains open in general.