Resonance phenomena have been the topic of a number of reviews, and separate questions have been elucidated in some monographs. But the absence of a bal anced integral account of the current status of the problem hinders the orientation in this area. The present book is an attempt to fill this gap. The results of investigations of the resonance scattering of electrons by atoms and ions are considered. We compare different theoretical methods of descrip tion of resonance phenomena, for example, the close-coupling method, R-matrix method, and diagonalization method. Special attention is paid to the analysis of the accuracy of the theoretical calculations and experimental data. Besides the conventional analytical solutions of a multiparticle problem, more recently developed methods, made possible by high speed computers, are discussed in detail. Several computer programs are scrutinized. This book is intended for physicists engaged in the problems of electronic and atomic collisions, and related areas such as plasma and laser physics. It should be of interest to university students and postgraduates.