When a liquid drop is placed on a smooth, rigid, solid substrate, it spreads until the final thermodynamic equilibrium is attained. The kinetics of spreading of the drop are controlled by conversion of capillary potential energy into viscous dissipation within the liquid. However, if the solid is sufficiently soft, a local deformation, or "wetting ridge", may form near the wetting front and the motion of the latter may lead to viscoelastic dissipation. We describe cases in which viscoelastic dissipation dominates and thus where spreading speed depends on bulk properties of the solid, rather than on liquid viscosity. This behavior in wetting can be considered to be analogous to dissipation phenomena in the adhesion of elastomers. Therefore, a comparison is made between wetting and adhesion dynamics. It appears clearly that a parallel can be drawn between the wetting of rubber and rubber adhesion to glass when both phenomena involve the same viscoelastic material. Kinetics of formation and breaking of corresponding interfaces is controlled by the damping properties of the soft solid substrate. In this paper, the viscoelastic properties of elastomers are described by two parameters, n and U-o; n is the usual speed power factor (n approximate to 0.5-0.6) and U-o a characteristic speed below which a fraction of the elastic strain energy in the wetting ridge is dissipated.