The principal goal of the book is to describe new accurate and robust algorithms for open resonant structures with substantially increased efficiency. These algorithms allow the extraction of complete information with estimated accuracy concerning the scattering of transient electromagnetic waves by complex objects. The determination and visualization of the electromagnetic fields, developed for realistic models, simplify and significantly speed up the solution to a wide class of fundamental and applied problems of electromagnetic field theory. The book presents a systematic approach to the study of electromagnetic waves scattering which can be introduced in undergraduate/postgraduate education in theoretical and applied radiophysics and different advanced engineering courses on antenna and wave-guide technology. On a broader level, the book should be of interest to scientists in optics, computational physics and applied mathematics.