This dissertation studies the logic behind quantum physics, using category theory as the principal tool and conceptual guide. To do so, principles of quantum mechanics are modeled categorically. These categorical quantum models are justified by an embedding into the category of Hilbert spaces, the traditional formalism of quantum physics. In particular, complex numbers emerge without having been prescribed explicitly. Interpreting logic in such categories results in orthomodular property lattices, and furthermore provides a natural setting to consider quantifiers. Finally, topos theory, incorporating categorical logic in a refined way, lets one study a quantum system as if it were classical, in particular leading to a novel mathematical notion of quantum-