This research report contains results obtained by the authors in recent years in the research area of dynamic fracture mechanics and wave propagation in damaged solids. It deals with several topics on wave propagation in elastic solids with cracks. Coverage includes wave scattering problems by a single crack, a periodic array of collinear cracks in isotropic and transversely isotropic elastic solids, interface cracks with a periodic spacing, and randomly distributed micro-cracks. The authors present a non-hypersingular boundary integral equation method to treat single-crack wave scattering problems, both in time - and frequency-domain. Numerical examples are given to show the accuracy and efficiency of the method.