We study the robustness of H∞ controllers, originally designed for nominal linear or nonlinear systems, to unknown static nonlinear perturbations in the state dynamics, the measurement equation, and the performance index. When the nominal system is linear, we consider both perfect state measurements and general imperfect state measurements, and in the case of nominally nonlinear systems, we consider perfect state measurements only. Using a differential game theoretic approach, we show for the former class that as the perturbation parameter (say, ε>0) approaches zero, the optimal disturbance attenuation level for the overall system converges to a value that is bounded above by the optimal disturbance attenuation level for the nominal system if the nonlinear structural uncertainties satisfy certain prescribed growth conditions. In particular, the optimal disturbance attenuation level for the overall system converges to the optimal disturbance attenuation level for the nominal system under perfect state measurements. We also show that the H∞-optimal controller designed based on a chosen performance level for the nominal linear system achieves the same performance level when the parameter |ε| is smaller than a computable threshold, except for the finite-horizon imperfect state measurements case. For the case, we show that the design of the nominal controller must be based on a decreased confidence level of the initial data, and a controller thus designed again achieves a desired performance level in the face of nonlinear perturbations satisfying a computable norm bound. In the case of nominally nonlinear systems, and assuming that the nominal system is solvable, we obtain sufficient conditions such that the nominal controller achieves a desired performance in the face of perturbations satisfying computable norm bounds. In this way, we provide a characterization of the class of uncertainties that is tolerable for a controller designed based on the nominal system.