[Truncated abstract] This thesis presents three different methods to investigate flexural wave propagation and scattering, power flow and transmission efficiencies, and dynamic stress concentration and fatigue failures in structural dynamics. The first method is based on the acoustical wave propagator (AWP) technique, which is the main part described in this thesis. Through the numerical implementation of the AWP, the complete information of the vibrating structure can be obtained including displacement, velocity, acceleration, bending moments, strain and stresses. The AWP technique has been applied to systems consisting of a one-dimensional stepped beam, a two-dimensional thin plate, a thin plate with a sharp change of section, a heterogeneous plate with multiple cylindrical patches, and a Mindlin?s plate with a reinforced rib. For this Mindlin?s plate structure, through the comparison of the results obtained by Mindlin?s thick plate theory and Kirchhoff?s classical thin plate theory, the difference of theoretical predicted results is investigated. As part of these investigations, reflection and transmission coefficients, power flow and transmission efficiencies in a onedimensional stepped beam, and power flow in a two-dimensional circular plate structure, are studied. In particular, this technique has been successfully extended to investigate wave propagation and scattering, and dynamic stress concentration at discontinuities. Potential applications are fatigue failure prediction and damage detection in complex structures. The second method is based on experimental techniques to investigate the structural response under impact loads, which consist of the waveform measuring technique in the time domain by using the WAVEVIEW software, and steady-state measurements by using the Polytec Laser Scanning Vibrometer (PLSV) in the frequency domain. The waveform measuring technique is introduced to obtain the waveform at different locations in the time domain. These experimental results can be used to verify the validity of predicted results obtained by the AWP technique. Furthermore, distributions of dynamic strain and stress in both near-field (close to discontinuities) and far-field regions are investigated for the study of the effects of the discontinuities on reflection and transmission coefficients in a one-dimensional stepped beam structure. Experimental results in the time domain can be easily transferred into those in the frequency domain by the fast Fourier transformation, and compared with those obtained by other researchers. This PLSV technique provides an accurate and efficient tool to investigate mode shape and power flow in some coupled structures, such as a ribbed plate. Through the finite differencing technique, autospectral and spatial of dynamic strain can be obtained. The third method considered uses the travelling wave solution method to solve reflection and transmission coefficients in a one-dimensional stepped beam structure in the time domain. In particular, analytical exact solutions of reflection and transmission coefficients under the given initial-value problem are derived. These analytical solutions together with experimental results can be used to compare with those obtained by the AWP technique.