Surface waves on smooth curved boundaries consist of an infinite number of modes, the lowest of which may be classified as of “Rayleigh type,” since it tends towards the Rayleigh wave in the limit of vanishing curvature. The others, which correspond to the “Whispering Gallery” modes of Lord Rayleigh, combine to form the lateral wave(s) in the zero-curvature case. For an elastic cylinder imbedded in a fluid, we have developed a computer program, combined with an analytic approach, in order to compute these modes and their dispersion curves. The results may be compared with asymptotic forms for a cylinder in a vacuum given by Brekhovskikh [Sov. Phys. Acoust. 13, 462 (1968)]. The mentioned surface waves play a role in scattering and target strength calculations, and these aspects of the problem are now being studied. [Supported in part by the Office of Naval Research.]