The paper considers a circular cylindrical three-layer shell of arbitrary thickness from a viscoelastic material. It is believed that it consists of two outermost bearing layers and a middle layer between them, the materials of which are generally different. The problem of unsteady torsional vibrations of such a shell with rigid contact between the layers is formulated. Proceeding from the assumption that there is a rigid contact between the layers, the dynamic and kinematic contact conditions of the problem are formulated. On the basis of exact solutions in transformations of the three-dimensional problem of the linear theory of viscoelasticity for a circular cylindrical three-layer shell, a mathematical model of its unsteady torsional vibrations has been developed. The proposed model includes the derivation of the general equations of torsional vibrations of the shell with respect to two auxiliary functions, which are the main parts of the torsional displacement of the points of some intermediate surface of the middle layer of the shell. Along with the equations, an algorithm for calculating was created that allows, based on the results of solving the equations of vibration, to unambiguously determine the stress-strain state of the shell and its layers in their arbitrary sections.