Methods of nonequilibrium thermodynamics and continuum mechanics are used for studying phase transitions of the first order in deformable solids with elastic and viscoelastic rheology. A phase transition of the first order is treated as the transition from one branch of the response functional to another as soon as state parameters reach certain threshold values determined by thermodynamic phase potentials and boundary conditions of the problem. Notions of kinematic and rheological characteristics of a phase transition associated with the change of the symmetry group due to the structural transformation and with the difference between thermodynamic potentials in undistorted phase configurations are introduced. In a quasi-thermostatic approximation, when inertia forces and temperature gradients are small, a close system of equations on the interface between deformable solid phases is formulated using laws of conservation. The system of the latter includes, in addition to the traditional balance equations of mass, momentum and energy, the divergence equation ensuring the compatibility of finite strains and velocities. As distinct from the classical case of the liquid gas phase equilibrium, the phase transition in solids is supposed to be irreversible due to the presence of singular sources of entropy of the delta function type whose carrier concentrates on the interface between the phases. The relations on the interface including the continuity conditions of the displacement vector, temperature, mass flux and the stress vector, as well as a certain restraint imposed on the jump of the normal component of the chemical potential tensor, are discussed. The latter restraint makes the resulting relations basically distinct from the classical conditions of the phase equilibrium. A generalized Clapeyron-Clausius equation governing the differential dependence of the phase transition temperature on the initial phase deformation is formulated. The paper presents a new relation of the phase transformation theory, namely, the equation describing the differential dependence of the phase transition temperature on the interface orientation relative to the anisotropy axes and the principal axes of the initial phase strain tensor. Based on the relations derived in this study, the phase transformation temperature of an initially isotropic material is shown to assume extreme values if the normal to the interface coincides with the direction of a principal axis of the initial phase strain tensor. The phase transition of the first order in a linear thermoelastic material with small strain values and small deviations of the temperature from its initial value is discussed in detail. A class of materials is distinguished in which an increase in the initial phase strain necessarily changes the character of the phase transformation a normal phase transition becomes an anomalous one and vice versa. The equilibrium of a compressed viscoelastic layer admitting melting and the effect of stress relaxation in the solid phase on the fluid boundary motion are examined.