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This book treats a selection of topics in electro-diffusion - a nonlinear transport process whose essence is diffusion of charged particles, combined with their migration in a self-consistent electric field. Basic equations of electro-diffusion were formulated about 100 years ago by Nernst and Planck in the ionic context. Sixty years later Van Roosbroeck applied these equations to treat the transport of holes and electrons in semiconductors. Correspondingly, major applications of the theory of electro-diffusion still lie in the realms of chemical and electrical engineering, related to its separation and semiconductor device technology. Some aspects of electro-diffusion are relevant for electrophysiology. Generally, this text has a dual orientation. On the one hand, it is addressed to persons whose research interests are directly related to electro-diffusion, i.e.,chemical engineers, membranologists, electrochemists, and electrophysiologists, with certain parts of this book possibly of interest to semiconductor device engineers. On the other hand, it aims to attract applied mathematicians to a practically relevant and remarkably underdeveloped classical branch of nonlinear mass-transfer. The entire presentation is largely heuristic with physical clarity prefered to mathematical generality. The main emphasis is placed upon examples that expose a few key situations. Formulations of unsolved problems are presented at the end of each chapter. it has attempted to make each chapter self contained, therefore, every chapter has a separate bibliography.