Solutions of Maxwell's equations are presented which satisfy the boundary conditions for corrugated and dielectric‐clad circular conducting cylinders. These solutions have the physical interpretation of leaky azimuthal surface waves. Values of the complex propagation constant are given as functions of the geometry. For large cylinders the leakage is small and the transmission properties are approximately those of a trapped wave on a flat surface. A coarse experiment gives reasonable agreement with the theory. Some possible applications to transmission lines and antennas are indicated.