Classical ring springs are mechanical elements used in industrial applications and in transport for shock absorption and energy dissipation. They are constituted by a stack of internal and external metal rings (typically high strength steel), with tapered surfaces in contact with one another. Under the action of an axial load these surfaces slide, the rings are deformed circumferentially and energy is dissipated due to friction. The main advantages of these springs are the high specific energy stored and the large damping capacity due to sliding friction. Furthermore, the stiffness and damping are independent on the strain rate and the temperature, which limits or avoids the occurrence of any resonance problems. The superelastic materials, characterized by an almost flat stress plateau and large reversible deformation, can be used to replace traditional steels in ring springs giving a significant performance increase. Compared to the traditional version where energy is dissipated only due to friction, in superelastic ring springs there is an increase of the dissipated energy thanks to the internal hysteresis of the material. This paper studies analytically the ring springs in traditional material and in superelastic material, providing equations to dimension these mechanical elements, which enable the designer to customize this useful structural element.