This book is an extensive revision of the earlier book with the same title, 1981. The reader who has the first edition will recognise chapters 1-3 and parts of Chapt. 4; thereafter, there is little similarity. There are several reasons for this. Firstly, as one internal Danish publication stated, the first edition contained "et hav af smafejl" meaning (with some poetic license), a sea of troubles. I welcomed the opportunity of setting this right. Secondly, a number of readers rightly criticised my glib dismissal of problems connected with homogeneous chemical reactions in the first edition, now dealt with. Thirdly, digital simulation has not stood still since 1980 and this book reflects the newer developments such as orthogonal collocation, algorithms for the implicit boundary value calculations, as well as my own recent findings. Despite all this activity, no text has appeared to displace that of Feldberg (1969) or my first edition of 1981. My aim is to explain the subject clearly and simply. The electrochemist (or student) who occasionally needs to simulate a diffusion problem numerically but does not wish to go in for sophisticated methods, can make do with the first four chapters. Desk time is then traded for computer time. If this becomes excessive and/or the simulator's interest in more efficient techniques is aroused, Chapt. 5 provides an introduction. The easiest of these is the Runge-Kutta integration method, providing a modest but definite gain in efficiency at very small cost in terms of desk time.