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In this paper, we review constitutive models for soft materials. We specifically focus on physically based models accounting for hyperelasticity, visco-hyperelasticity, and damage phenomena. For completeness, we include the thermodynamically based viscohyperelastic and damage models as well as the so-called mixed models. The models are put in the frame of statistical mechanics and thermodynamics. Based on the available experimental data, we provide a quantitative comparison of the hyperelastic models. This information can be used as guidance in the selection of suitable constitutive models. Next, we consider visco-hyperelasticity in the frame of the thermodynamic theory and molecular chain dynamics. We provide a concise summary of the viscohyperelastic models including specific strain energy density function, the evolution laws of internal variables, and applicable conditions. Finally, we review …