In view of the large amount of recent work on the thermodynamics of non-equilibrium processes, there is a real need for a book giving a clear exposition of the thermodynamic formalism applicable to nonlinear thermodynamic processes at a mathematical level, accessible to physicists and to theoretically inclined biologists and chemists. This book attempts to fill this need by making a definite statement in regard to the present-day status of linear and nonlinear thermodynamics.