  The first part of this paper contains an introduction to Bell inequalities
and Tsirelson's theorem for the non-specialist. The next part gives an explicit
optimum construction for the "hard" part of Tsirelson's theorem. In the final
part we describe how upper bounds on the maximal quantum violation of Bell
inequalities can be obtained by an extension of Tsirelson's theorem, and survey
very recent results on how exact bounds may be obtained by solving an infinite
series of semidefinite programs.
