  The notion of contextuality, which emerges from a theorem established by
Simon Kochen and Ernst Specker (1960-1967) and by John Bell (1964-1966), is
certainly one of the most fundamental aspects of quantum weirdness. If it is a
questioning on scholastic philosophy and a study of contrafactual logic that
led Specker to his demonstration with Kochen, it was a criticism of von
Neumann's "proof" that led John Bell to the result. A misinterpretation of this
famous "proof" will lead them to diametrically opposite conclusions. Over the
last decades, remarkable theoretical progresses have been made on the subject
in the context of the study of quantum foundations and quantum information.
Thus, the graphic generalizations of Cabello-Severini-Winter and
Acin-Fritz-Leverrier-Sainz raise the question of the connection between
non-locality and contextuality. It is also the case of the sheaf-theoretic
approach of Samson Abramsky et al., which also invites us to compare
contextuality with the logical structure of certain classical logical
paradoxes. Another approach, initiated by Robert Spekkens, generalizes the
concept to any type of experimental procedure. This new form of "universal"
contextuality has been raised as a criterion of non-classicality, i.e. of
weirdness. It notably led to identify the nature of curious quantum paradoxes
involving post-selections and weak measurements. In the light of the fiftieth
anniversary of the publication of the Kochen-Specker theorem, this report aims
to introduce these results little known to the French scientific public, in the
context of an investigation on the nature of the weirdness of quantum physics.
