The nonlinear dynamics of a single-degree-of-freedom pneumatic vibration isolator are studied. Based on a physical model, a nonsymmetric stiffness nonlinearity is derived to describe the stiffness property of the isolator. For a full nonlinear pneumatic isolator model, the response to two different types of disturbances is studied: forces applied to the isolated payload and base vibrations. The dynamic behavior of the isolator in case of a disturbance applied to the payload is studied using the generalized force-mobility function and features coexisting steady-state responses and a superharmonic resonance. Base vibrations transmitted via the isolator are studied on the basis of the generalized transmissibility function again showing rich nonlinear dynamic behavior. The presence of a nonsymmetric nonlinearity also induces high-energy low-frequency response to multiple high-frequency excitation. For both types of excitation, the nonlinear behavior is seriously compromising the performance of the isolator. To avoid any expression of nonlinearity whatsoever and, at the same time, to enhance the performance of the passive isolator, an overall nonlinear control design is proposed. It consists of a linear PID-based controller together with a nonlinear computed torque controller (CTC). For either linear or nonlinear control, the isolator performance is quantified in terms of generalized force mobility and transmissibility. The latter with a special focus on multiple high-frequency excitation.