This book is an expanded revision of the second edition of Grober/Err, which was published under the same title in 1933 by Springer-Verlag. The task faced by the editor of the new edition varied significantly for the individual parts of the book. The first part, concerning heat conduction in solid bodies written by Grober, primarily dealt with the solution methods of the Fourier equation to an extent that was sufficient for engineering practice in most cases. Grober's particular merit lies in having found an accessible and comprehensible presentation of the analytical theory of heat conduction for engineers, much of which originated from the first edition of the book (1921). Thus, the Grober/Err has indeed served an entire generation of engineers as a textbook on the theory of the Fourier equation.