FBstab: a proximally stabilized semismooth algorithm for convex quadratic programming. This paper introduces the proximally stabilized Fischer-Burmeister method (FBstab); a new algorithm for convex quadratic programming that synergistically combines the proximal point algorithm with a primal-dual semismooth Newton-type method. FBstab is numerically robust, easy to warmstart, handles degenerate primal-dual solutions, detects infeasibility/unboundedness and requires only that the Hessian matrix be positive semidefinite. We outline the algorithm, provide convergence and convergence rate proofs, and report some numerical results from model predictive control benchmarks and from the Maros-Meszaros test set. We show that FBstab is competitive with state of the art methods and is especially promising for model predictive control and other parameterized problems.