It is well-known that the Omitting Types Theorem from model theory can be proved by topological means, and the central ingredient of that proof is the Baire Category Theorem. The goal of this talk is to consider the extent to which the Omitting Types Theorem is equivalent to the Baire Category Theorem. To do so, we will describe a topological framework (based on work of Robin Knight) that generalizes the classical type spaces from model theory. Many classical logics (including first-order, infinitary, and continuous logics) fit into this general setting, and conversely we will show that each instance of the general framework yields a model-theoretic logic. We then distinguish several version of the Omitting Types Theorem these logics may have, based on Baire Category properties of the underlying topological spaces. All of these properties are equivalent for first-order logic, but are distinct in the general setting. This is joint work with Frank Tall.