We investigate the approximability of the following optimization problem, whose input is ann-by-n matrix A and an origin symmetric convex set C that is given by a membership oracle:"Maximize the quadratic form as x ranges over C."This is a rich and expressive family of optimization problems; for different choices of forms Aand convex bodies C it includes a diverse range of interesting combinatorial and continuousoptimization problems. To name some examples, max-cut, Grothendieck's inequality, thenon-commutative Grothendieck inequality, certifying hypercontractivity, small set expansion,vertex expansion, and the spread constant of a metric, are all captured by this class. While theliterature studied these special cases using case-specific reasoning, here we develop a generalmethodology for treatment of the approximability and inapproximability aspects of these questions.Based on joint work with Euiwoong Lee and Assaf Naor.