The results for vibrations of an elastic cylinder as predicted by a number of the approximate shell theories are compared with the results of the exact theory. It is found that the membrane theory of shells is accurate for predicting frequencies and displacement ratios of cylinders with appreciable thickness. Furthermore, the theories which include bending effects as well as membrane effects are good at even the shorter wavelengths; and the theories which include rotatory inertia and shear are accurate over most of the wavelength spectrum of the lowest branch of rather thick shells. However, for the very thick shell (with a ratio of inside radius to outside radius less than 0.5), only the exact theory shows the full characteristics of the displacement distribution.