  When it isn't possible to tell two distinct experimental procedures apart
purely from their input/output statistics, then it seems a plausible hypothesis
that the two procedures must be physically identical. We call such a hypothesis
"noncontextuality", an instance of Leibniz's principle of the identity of
indiscernibles. Read in the contrapositive, this hypothesis entails that any
physical distinctions between two experimental procedures must necessarily lead
to a difference in their operational statistics. The results I present in this
thesis concern the failure of this hypothesis -- a failure dubbed
"contextuality" -- when one tries to embed an operational theory (such as
quantum theory) in the ontological models framework. The Kochen-Specker theorem
demonstrates the failure of noncontextuality for deterministic ontological
models of quantum theory, i.e., those ontological models where the
ontic/physical state of the system fixes the outcome of any projective
measurement on the system in a deterministic manner. This thesis goes beyond
the Kochen-Specker (KS) theorem by asking what operational facts must be
verified in experiments to conclude that Nature does not admit noncontextual
ontological models, not even indeterministic ones. This leads to
noncontextuality inequalities that are robust to noise in the preparations and
measurements. In the particular case of quantum theory, these inequalities are
meaningful even when unsharp measurements (or POVMs) are allowed, a feature not
shared by the traditional approach to KS-noncontextuality where unsharp
measurements are excluded by fiat: allowing them renders even trivial POVMs
(proportional to identity) maximally KS-contextual. The sense in which trivial
POVMs are indeed "trivial" (or "noncontextual") is clear in our approach: they
are simply too noisy to lead to violation of our noncontextuality inequalities.
