Katsenelenbaum, retired from the Moscow Institute of Physics and Technology, investigates the properties of fields scattered by metallic bodies and examines their significance for antenna synthesis, body shape reconstruction, and other diffraction problems. The material covered lies at the interface of antenna theory, high-frequency electrodynamics, mathematical physics, and inverse scattering problems, and will be of interest to researchers working in electromagnetic waves and diffraction theory. The mathematical technique, according to the author, "is not more difficult than that used in diffraction theory problems." The book was first published in Russian (Nauka, Moscow, 1996) under the title The Approximation Problem of Electromagnetic Fields. This English edition is extended with new results and contains a new chapter on long narrow beam of electromagnetic waves, and a new appendix on antenna synthesis by the amplitude radiation pattern and the modified phase problem.