Covers key theoretical concepts and applications of wave propagation with the focus on ultrasonic scattering. The course material includes: (i) dispersion analysis for design and characterization of materials with random and periodic microstructure, and (ii) advanced elastodynamic formulations rooted in 2D and 3D boundary integral equations for the purpose of inverse scattering and imaging. The course is intended to be self-contained, research-oriented and accessible for multidisciplinary audience.