The propagation of high frequency elastic surface waves along the generators of a homogeneous isotropic cylinder which has a cross-sectional boundary of nonconstant curvature is investigated. The boundary surface is stress-free and the surface waves, or Rayleigh waves, are disturbances whose amplitudes decay rapidly with depth into the cylinder. In the case of an open boundary curve for which the curvature attains its algebraic maximum at a single point, it is found that modes exist for which the disturbance is essentially confined to a region in the neighborhood of the point of maximum curvature, as well as to the neighborhood of the surface. The amplitude of the disturbance decays rapidly on either side of the point of maximum curvature. The application of these high frequency asymptotic results to the case of a closed boundary curve is discussed. Particular cases will be investigated in more detail in a subsequent paper.