<p>The review is proposed with the investigations related to the dynamics of the oscillating, moving, and oscillating-moving loads acting on the inner surface of the hollow cylinder surrounded by the elastic medium. All considered investigations are carried out within the framework of the piecewise homogeneous body model by utilizing the exact 3D equations and relations of the linear and linearized elastodynamics. The problems are classified as axisymmetric and 3D non-axisymmetric, as well as the axisymmetric problems related to the case where the constituents of the system have non-homogeneous initial stresses. For further investigations, the subjects related to the dynamics of the moving and oscillating moving load acting in the interior of the hollow cylinder surrounded by the elastic and viscoelastic medium are suggested.</p>