Due to the continued research into chemistry, biology, pharmaceutics and rocket space technology, interest in the study of the dynamics of layered fluids has increased significantly. The paper focuses on oscillations of a three-layer viscous fluid, gives the formulation of the viscous fluid free oscillations problem. Within the research, we determined natural frequencies and damping coefficients of oscillations of the three-layer viscous fluid in a cylindrical vessel by means of the boundary layer method and a mechanical analog. Oscillations of the three-layer viscous fluid were considered as joint oscillations of two partial hydrodynamic systems, one of which corresponds to oscillations of the upper and middle viscous fluids, and the other one - to oscillations of the middle and lower fluids. Then, we determined the coefficients of viscous resistance in partial hydrodynamic systems of a two-layer viscous fluid. Using the mechanical analog of oscillations of the three-layer liquid, we derived the characteristic equation for determining natural frequencies of the hydrodynamic system under consideration. Next, we calculated the dependency of natural frequencies and liquid-liquid interface damping coefficients on the height of the middle layer and the density of the upper fluid. Finally, we analyzed and compared theoretical calculations with the results obtained by other researchers and experimental investigation. The paper gives the results of experimental studies of oscillations of the three-layer fluid in a stationary cylindrical tank.