I will discuss a recent paper of mine, the aim of which is to count the number of prime solutions to Q(p_1,..,p_8) = N, for a fixed quadratic form Q and varying N. The traditional approach to problems of this type, the Hardy-Littlewood circle method, does not quite suffice. The main new idea is to involve the Weil representation of the symplectic groups Sp_8(Z/qZ). I will explain what this is, and what it has to do with the original problem. I hope to make the talk accessible to a fairly general audience.