Finite element analysis of the time-dependent deformations of layered viscous solids serves as the basis of the study of the mechanics of folding. The progressive development of folds by buckling in single and multilayer models compressed parallel to the layering is reviewed. Fold geometries are shown to vary from parallel, for large viscous contrasts, to nearly similar, for low contrasts. For models with the same viscosity contrast the geometry depends upon the wavelength/thickness ratio, so that thin-layer folds behave in the most 'competent' fashion with a great amount of buckle shortening. The development of stresses around folds is discussed. As the fold grows the principal stresses rotate and the magnitude changes quite drastically for models with high viscosity contrast. These folds also have the gradient of mean stress directed perpendicular to the layer in the hinge part of the competent layer. The heterogeneous stress distribution, as it appears in a fold structure, generates a free energy gradient, and diffusion current will tend to bring the system to a state of equilibrium by one or more of the following events: (1) introduction of new mineral species; (2) polymorphic phase changes; (3) a change in chemical composition and (4) a change in grain size. Future development of the finite element analysis of folding is discussed.