This dissertation investigates the nonlinear wave propagation phenomena in nonlinear metamaterials with nonlinear chains and nonlinear resonators using analytical and spectro-spatial analyses. In the first part of the thesis, the nonlinear metamaterials are modeled as a chain of masses with multiple local resonators attached to each cell. The nonlinearity stems from the chain's stiffness in one case and the local resonator's stiffness in another. Analytical approximates solutions are obtained for each case using perturbation techniques. These results are validated through numerical simulations and the results show good agreement. To further demonstrate the nonlinear wave propagation characteristics, spectro-spatial analyses are conducted on the numerical integration data sets. The wave profiles, short-term Fourier transform spectrograms, and contour plots of 2D Fourier transform show the presence of solitary waves for both sources of nonlinearity. In addition, spectro-spatial features demonstrate the presence of significant frequency shifts at different wavelength limits. indent The second part of the thesis studies a nonlinear electromechanical metamaterial and examines how the electromechanical coupling in the local resonator affects the wave propagation. Numerical examples indicate that the system can be used for simultaneous energy harvesting and vibration attenuation without any degradation in the size of bandgaps. Spectro-spatial analyses conducted on the electromechanical metamaterial also reveal the presence of solitons and frequency shifts. The presence of solitary wave in the electromechanical metamaterial suggests a significant improvement in energy harvesting and sensing techniques. The obtained significant frequency shift is employed to design an electromechanical diode, allowing voltage to be sensed and harvested only in one direction. Design guidelines and the role of different key parameters are presented to help designers to select the type of nonlinearity and the system parameters to improve the performance of acoustic diodes. indent The last part of this thesis studies the passive self-tuning of a metastructure via a beam-sliding mass concept. The governing equations of motions of the holding structure, resonator, and sliding mass are presented and discretized into a system of ODEs using Galerkin's projection. Given that the spatial parameters of the system continuously change over time (i.e., mode shapes and frequencies), instantaneous exact mode shapes and frequencies are determined for all possible slider positions. The numerical integration is conducted by continuously updating the spatial state of the system. The obtained exact mode shapes demonstrate that the resonance frequency of the resonator stretches over a wide frequency band. This observation indicates that the resonator can attenuates vibrations at a wide frequency range. Experiments are also conducted to demonstrate the passive self-tunability of the metastructure and the findings colloborate the analytical results. / Doctor of Philosophy / Metamaterials are artificially engineered structures that can offer incredible dynamical properties, which cannot be found in conventional homogeneous structures. Consequently, the global metamaterials market is expected to display a 23.6$%$ compound annual growth rate through 2027. Some of these exciting properties include, but not limited to, negative stiffness, negative mass, negative Poisson's ratio. The unique dynamic properties show the importance of metamaterials in many engineering applications, such as vibration reduction, noise control, and waveguiding and localization. However, beyond the linear characteristics of metamaterials, nonlinear metamaterials can exhibit more interesting nonlinear wave propagation phenomena, such as solitons, cloaking, tunable bandgaps, and wave non-reciprocity. indent This research work investigates wave propagation characteristics in nonlinear locally resonant metamaterials using analytical, numerical, and signal processing techniques. The nonlinearity stems from the chain in one case and from the local resonator in another. Numerical examples show the presence of solitary waves in both types of nonlinearity and significant frequency shift in certain frequency/wavenumber regions. The obtained significant frequency shift can be utilized to design mechanical diodes, where its operation range can be increased by introducing nonlinearity in the resonator. indent For simultaneous energy harvesting and vibration attenuation, integrating the local resonator with piezoelectric energy harvesters is also investigated in this research work with the presence of both types of nonlinearities. For weak electromechanical coupling, the results demonstrate that the band structure of the system is not affected by the electromechanical coupling. Therefore, the system can also be used for energy harvesting without any degradation in the vibration attenuation performance. This observation is also validated experimentally for the linear limit. Spectro-spatial analyses also reveal the presence of solitary output voltage waves, which can enhance the energy harvesting and sensing. The obtained significant frequency shift can be utilized to design an electromechanical diode where the wave can propagate and be harvested only in one direction. Numerical examples show that the performance of the electromechanical diode can be significantly improved by including nonlinearities in the local resonator. indent Another goal of this research work is the introduction of passive self-tuning mechanism to design self-tuning metastructure. The design of such a metastructure is motivated by the need for broadband devices that can adapt to changing environment. The passive self-tuning concept is achieved by a sliding mass coupled with a resonator. Analytical and experimental results show the ability of this system to tune itself to the excitation frequency, and hence, can control vibrations over a significantly wider frequency band as compared to conventional resonators.