A non-linear diffraction theory of periodic gravity waves due to their interaction with a circular cylinder which extends from the ocean bottom through the free surface is presented. In order to obtain the solution, the velocity potential, the free surface elevation and the frequency parameter are expanded as series in powers of a parameter which is the ratio of the amplitude to wave length. Knowing the velocity potential of the incident wave, the velocity potential of the scattered wave for the corresponding order is obtained satisfying the proper boundary conditions and the radiation condition. The total potential governing the motion is thus obtained as a sum of incident and scattered potential for computing the pressures and forces on the cylinder. © 1975 Springer-Verlag.