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In this thesis we explore the questions of what should be considered a"classical" theory, and which aspects of quantum theory cannot be captured byany theory that respects our intuition of classicality.  This exploration is divided in two parts: in the first we review classicalresults of the literature, such as the Kochen-Specker theorem, von Neumann'stheorem, Gleason's theorem, as well as more recent ideas, such as thedistinction between $\psi$-ontic and $\psi$-epistemic ontological models,Spekkens' definition of contextuality, Hardy's ontological excess baggagetheorem and the PBR theorem.  The second part is concerned with pinning down what should be the "correct"definition of contextuality. We settle down on the definition advocated byAbramsky and Branderburger, motivated by the Fine theorem, and show theconnection of this definition with the work of George Boole. This definitionallows us to unify the notions of locality and noncontextuality, and uselargely the same tools to characterize how quantum mechanics violates thesenotions of classicality. Exploring this formalism, we find a new family ofnoncontextuality inequalities. We conclude by reviewing the notion ofstate-independent contextuality.

Quantum realism and quantum surrealism