Nonlinear systems leveraging the effects of negative stiffness can exhibit beneficial qualities for passive seismic mitigation in structures. Such systems can be achieved by placing nonlinear devices displaying negative stiffness in parallel with linear positive stiffness systems such as a structure or spring. This thesis presents research into two such systems: (i) a device which causes apparent weakening in a structure subjected to horizontal ground motions and (ii) an isolation system to protect building contents from vertical seismic effects. Apparent weakening is the softening of a structure’s apparent stiffness by adding negative stiffness to the overall system via negative stiffness devices. Apparent weakening is an elastic effect that has the benefit of reducing the peak accelerations and base shears induced in a structure due to a seismic event without reducing the main structural strength. The smooth negative stiffness device (SNSD) presented in this thesis consists of cables, pulleys, and extension springs. A nonlinear mathematical model of the load-deflection behavior of the SNSD was developed and used to determine the optimal geometry for such a device. A prototype device was designed and fabricated for installation in a bench-scale experimental structure, which was characterized through static and dynamic tests. A numerical study was also conducted on two other SNSD configurations designed to achieve different load-deflection relations for use in an inelastic model building subject to a suite of historic and synthetic ground motions. In both the experimental prototype and the numerical study, the SNSDs successfully produced apparent weakening, effectively reducing accelerations and base shears of the structures. The buckled-strut vertical isolation system (BSVIS) presented in this thesis combines the non-linear behavior of a laterally-loaded buckled strut with a linear spring. The lateral load-deflection relation for a buckled strut, which is nonlinear and displays negative stiffness, was investigated for various conditions to two- and three-term approximations of the deflected shape of a strut. This relation and the linear positive effect of a spring were superimposed to give the load-deflection relation of a BSVIS. An experimental prototype was fabricated and subjected to static tests. These tests confirmed the validity of the model and the effectiveness of adding a spring in parallel with a buckled strut to achieve isolation-level stiffness. Based on the theoretical and experimental findings, a design guide is proposed for the engineering of a BSVIS to protect a payload from vertical seismic content.