Soft matter has become one of the most active fields since the 1990 s, for it has enormous interesting behaviors and a broad range of applications. Rational continuum mechanics, as a subject mainly dealing with the kinematics and deformation of materials modeled as continuous mass, is a main source of inspiration in the development of soft matter physics. Here we review the development of rational continuum mechanics and soft matter briefly, and focus on the basic mechanical models and constitutive relations relating to soft matter: entropy elasticity, hyperelasticity, viscoelasticity, poroelasticity, non-Newtonian fluid, and the constitutive equations of these models. We simultaneously introduce the applications of these equations in hot issues in recent years, such as brain, blood vessel, cartilage, muscle, gel, cell, three dimensional printing, etc. According to applications and advances in soft matter mechanics, we then propose the key scientific problems and research fronts: mechanics of the solid-liquid interfacial interactions, introducing multiple factors into constitutive equations to describe the complex behaviors of soft matter in coupling multi-physics, and enhancing connections between soft matter mechanics and soft matter physics, chemistry, biology, etc. Finally, we conclude that the rational continuum mechanics in soft matter could be further developed in energy development, fabrication and analysis of diverse soft materials, and biomedicine development areas.