We obtain the characteristic equation of an elastic cylinder, or cylindrical shell, immersed in a fluid and enclosing another fluid for vibrations containing both azimuthal and axial components. Numerical solutions are found for aluminum and steel cylinders, in the form of distinct “layers” of roots in the complex frequency plane, which may be interpreted as the resonances of azimuthal or helical surface waves around the shell. Rayleigh waves, Stoneley waves, and Lamb (whispering gallery) waves are identified and the interaction of Rayleigh and Stoneley branches at low frequency is clarified. Surface wave speeds and attenuations are obtained as functions of frequency. [Work supported by the Office of Naval Research.]