This volume provides a self-contained and systematic development of an aspect of analysis which deals with the theory of fundamental solutions for differential operators, as well as their applications to boundary value problems of mathematical physics, applied mathematics, and engineering. Features of the text include: extensive applications topics presented in detail, with worked examples; coverage of over 70 different differential operators and derivation of fundamental solutions for them by using Fourier transforms and the theory of distributions; computational components discussed in all relevant topics and applications; an appendix describing numerous algorithms, and programs in C to illustrate solutions methods. All computer-based material is available by FTP from the Birkhauser web site.