New models of viscoelastic solids at small and finite deformations are proposed that describe material failure by enforcing the energy limiter-the average bond energy. Basically, the bond energy defines the energy that is necessary to separate two attracting particles. In the case of a solid composed of many particles there exists a magnitude of the average bond energy that is necessary to separate particles in a small material volume. The average bond energy can be calculated if a statistical distribution of the bond density is known for a particular material. Alternatively, the average bond energy can be determined in macroscopic experiments if the energy limiter is introduced in a material constitutive model. Traditional viscoelastic models of materials do not have energy limiters and, consequently, they allow for unlimited energy accumulation under the strain increase. The latter is unphysical, of course, because no material can sustain large enough deformations without failure. The average bond energy is the energy limiter that controls material softening, which indicates failure. Thus, by limiting the stored energy we include a description of material failure in the constitutive model. Viscoelasticity including energy limiters can be called softening hyperviscoelasticity. We present two softening hyperviscoelasticity models for small and finite deformations. In all cases the elastic and viscoelastic responses are described by potentials with limiters, which control material softening. The models are studied in the case of simple shear and uniaxial tension. The results of the calculations show that softening hyperviscoelasticity can be used for analysis of rate-dependent failure of materials.