This work offers a comprehensive presentation of the mathematical-theoretical fundamentals of nondestructive material testing with ultrasound. Starting from the basic equations of elastodynamics, long and transverse waves as well as Gaussian beams in isotropic and anisotropic materials are discussed. The mathematical concept of Green's functions is then used to calculate probe sound fields and scattering fields of discontinuities. A broad scope is given to imaging ultrasonic methods. Numerous wave field simulations are used for illustrative purposes. About the authors: Prof. Dr. rer. nat. Karl-Jörg Langenberg has been teaching theoretical electrical engineering at the University of Kassel since 1983. Priv.-Doz. Dr.-Ing. René Marklein, senior engineer in the Department of Theory of Electrical Engineering and Photonics, University of Kassel; Berthold Award Winner of the DGZfP 1994. Dr.-Ing. Klaus Mayer, Akademischer Oberrat.