Various systems and structures are subjected to impact loading in industrial and military applications. Many of these impact loads have very high magnitudes and very short durations, resulting in high frequency content. Under some conditions, the response to these loading conditions can be significantly influenced by nonlinearities. The goal of this thesis is to develop new tools for studying the nonlinear wave propagation which can result from this extreme impact loading and provide an in-depth understanding of the underlying physical process. It consists of analytical, numerical, and experimental studies. Two new numerical methods are developed for high fidelity simulation of nonlinear wave propagations: the alternating frequency-time finite element method (AFT-FEM) and the alternating wavelet-time finite element method (AWT-FEM). A perturbation based approach is developed to derive analytical formula of the wavenumber for one-dimensional rod model. By employing these numerical and analytical methods, numerical simulations of wave propagations in both infinite and finite domains for one-dimensional and two-dimensional structures are conducted to explore nonlinear behaviors in the responses. Experimental efforts are also made to verify numerical results of impact wave propagation. Through comparison with other existing numerical approaches, the advantages of AWT-FEM in computational efficiency and high fidelity are demonstrated and the method is employed for applications of nonlinear force identification and drill-string stability monitoring.