This volume describes the application of the method of the differential specific forces (MDSF). By using this new method, the solutions to the problems of a dissipative viscoelastic and elastic-plastic contacts between curvilinear surfaces of two solid bodies can be found. The novelty is that the forces of viscosity and the forces of elasticity can be found by an integration of the differential specific forces acting inside an elementary volume of the contact zone. This volume shows that this method allows finding the viscoelastic forces for any theoretical or experimental dependencies between the distance of mutual approach of two curvilinear surfaces and the radiuses of the contact area. Also, the derivation of the integral equations of the viscoelastic forces has been given and the equations for the contact pressure have been obtained. The viscoelastic and elastic-plastic contacts at impact between two spherical bodies have been examined. The equations for work and energy in the phases of compression and restitution and at the rolling shear have been obtained. Approximate solutions for the differential equations of movement (displacement) by using the method of equivalent work have been calculated. This new method of differential specific viscoelastic forces allows us to find the equations for all viscoelastic forces. It is principally different from other methods that use Hertz’s theory, the classical theory of elasticity and the tensor algebra. This method will be useful in research of contact dynamics of any shape of contacting surfaces. It also can be used for determination of the dynamic mechanical properties of materials and in the design of wear-resistant elements and coverings for components of machines and equipment that are in harsh conditions where they are subjected to the action of flow or jet abrasive particles. This volume will be useful for professional designers of machines and mechanisms as well as for the design and development of new advanced materials, such as wear-resistant elastic coatings and elements for pneumatic and hydraulic systems, stop valves, fans, centrifugal pumps, injectors, valves, gate valves, and in other installations.