Over the last hundred years, the circle method has become one of the most important tools of analytic number theory. This talk (on joint work with Roger Heath-Brown) will describe a new application of the circle method to pairs of quadratic forms, via a novel two-dimensional analogue of Kloosterman's version of the circle method. As a result, we prove (under a mild geometric constraint) that any two quadratic forms with integer coefficients, in 5 variables or more, simultaneously attain prime values infinitely often.