We consider the optimal regulation problem for nonlinear control-affine dynamical systems. Whereas the linear-quadratic regulator (LQR) considers optimal control of a linear system with quadratic cost function, we study polynomial systems with polynomial cost functions; we call this problem the polynomial-polynomial regulator (PPR). The resulting polynomial feedback laws provide two potential improvements over linear feedback laws: 1) they more accurately approximate the optimal control law, resulting in lower control costs, and 2) for some problems they can provide a larger region of stabilization. In this talk, I will present some of our recent work developing scalable numerical methods for computing these polynomial feedback laws. Their performance will be illustrated first on a low-dimensional aircraft stall stabilization example, for which PPR control recovers the aircraft from more severe stall conditions than LQR control, and then on a semidiscretization of a partial differential equation.