A new class of adaptive passive stiffness systems undergoing nonlinear oscillations are proposed and studied in this Ph.D. thesis. Adaptive passive stiffness systems that have been proposed in this thesis and by other researchers recently allow for new ways of vibration isolation, seismic protection, and tuned mass damping of vibrations. New characteristics which emerge in these adaptive passive stiffness systems, like the coexistence of negative stiffness and positive stiffness, pose the need of analytical methods for studying these systems. Existing analytical and computational techniques that permit such a study are explored and new techniques are developed. A new analytical technique called the multi-frequency homotopy analysis method is proposed to this end. This multi-frequency homotopy analysis method can not only be adopted to solve a wide variety of nonlinear problems with periodic and quasi-periodic steadystate solutions, but can also be utilized to obtain analytical expressions of nonlinear dynamic systems’ transient response. These abilities of the multi-frequency homotopy analysis method make it a powerful tool for studying the new adaptive passive stiffness systems. Besides, the pseudo-force method is adapted to analytically solve a nonlinear system involving negative stiffness. Other analytical and numerical methods like the modified Lindstedt–Poincar´e method, the event-driven simulation technique and the time stepping method are also explored for analyzing such adaptive passive stiffness systems. In this Ph.D. study, a new adaptive passive stiffness device called universal stiffness devices (USD) is proposed. The newly proposed USD can generate a wide range of smooth and piecewise smooth restoring forces with positive as well as negative stiffnesses. The stiffness of the USD can be changed smoothly and continuously by simply adjusting a physical parameter of the USD. The variability of the USD’s stiffness can provide retuning capability of vibration isolation systems and other control systems with such a USD. The proposed USD can serve as a core element in a vibration isolation system to adaptively attenuate the vibration transmitted to the primary structure. Moreover, as an adaptive passive device, the USD brings better adaptability than a purely passive system. An analytical model of the USD has been established and experimental study has been conducted. The force–displacement loops of the USD are measured and compared to the results computed by using the newly formulated analytical model. The USD’s capability of changing its stiffness as it is designed has been tested. The performance of a vibration isolation system with the proposed USD has been verified experimentally under a range of harmonic excitations that cover dominant vibration frequencies and amplitudes. For future research, further application of the proposed analytical method for investigating nonlinear vibration isolation systems of different types could be expected; more practical applications of the proposed USD in vibration control and hazard reduction can be explored.