Passive vibration isolation may be a cost-effective solution to isolate a supported system containing a source and/or receiver from the supporting structure. The standard linear theory suggests a low-stiffness joint to create a mobility mismatch in the transmission path, but this solution may lead to large amplitude motions in the supported system. To achieve both motion control and isolation with the same mount and without compromising either objective, an innovative, nonlinear mount concept is proposed. Taking advantage of geometric nonlinearity for large displacements, a quasi-zero stiffness is generated by exploiting the interaction between the nonlinear mechanisms that govern the motion of a number of inclined shear legs. For example, a three-regime stiffness profile is created, including a medium-stiffness preload regime, a quasi-zero stiffness isolation regime, and a high-stiffness motion control regime. This concept offers significant benefits compared with a more conventional compromise approach in that low-amplitude vibrations are exceptionally isolated while large amplitude transient motions are controlled. Illustrative computational examples will be presented to support the underlying linear and nonlinear design principles. Limiting cases will be discussed as well.