This inequality is a fundamental and unifying tool in many areas of mathematics and computer science (functional analysis, combinatorics, machine learning, system theory, quantum information theory, numerical linear algebra, statistical mechanics, computational complexity). With hindsight one can view Grothendieck's inequality and its proof, which is algorithmic, as the first randomized approximation algorithm based on semidefinite programming.
In this talk I want to survey recent developments around Grothendieck-type inequalities.
Some relevant links:
 Mark Braverman, Konstantin Makarychev, Yury Makarychev, Assaf Naor - The Grothendieck constant is strictly smaller than Krivine's bound (http://arxiv.org/abs/1103.6161)
 Jop Briet, Fernando Mario de Oliveira Filho, Frank Vallentin - Grothendieck inequalities for semidefinite programs with rank constraint (http://arxiv.org/abs/1011.1754)
 Subhash Khot, Assaf Naor - Grothendieck-type inequalities in combinatorial optimization (http://arxiv.org/abs/1108.2464)