  This review article is concerned with a recently uncovered connection between
operator spaces, a noncommutative extension of Banach spaces, and quantum
nonlocality, a striking phenomenon which underlies many of the applications of
quantum mechanics to information theory, cryptography and algorithms. Using the
framework of nonlocal games, we relate measures of the nonlocality of quantum
mechanics to certain norms in the Banach and operator space categories. We
survey recent results that exploit this connection to derive large violations
of Bell inequalities, study the complexity of the classical and quantum values
of games and their relation to Grothendieck inequalities, and quantify the
nonlocality of different classes of entangled states.
