This book is an extensive revision of the earlier 2nd Edition with the same title, of 1988. The book has been rewritten in, I hope, a much more did- tic manner. Subjects such as discretisations or methods for solving ordinary di?erential equations are prepared carefully in early chapters, and assumed in later chapters, so that there is clearer focus on the methods for partial di?erential equations. There are many new examples, and all programs are inFortran90/95,whichallows amuchclearerprogrammingstylethanearlier Fortran versions. In the years since the 2nd Edition, much has happened in electrochemical digital simulation. Problems that ten years ago seemed insurmountable have been solved, such as the thin reaction layer formed by very fast homogeneous reactions, or sets of coupled reactions. Two-dimensional simulations are now commonplace, and with the help of unequal intervals, conformal maps and sparse matrix methods, these too can be solved within a reasonable time. Techniques have been developed that make simulation much more e?cient, so that accurate results can be achieved in a short computing time. Stable higher-order methods have been adapted to the electrochemical context. The book is accompanied (on the webpage www.springerlink.com/ openurl.asp?genre=issue&issn=1616-6361&volume=666) by a number of - ample procedures and programs, all in Fortran 90/95. These have all been veri?edasfaraspossible.Whilesomeerrorsmightremain,theyarehopefully very few.