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Bi-periodic homogeneous cylindrical structures are studied, for the first time, under a waveguide viewpoint. The periodic nature of the structure allows for its characterization by means of a single period, or unit cell. Accordingly, the eigenvalues and eigenfunctions of the wave equation, under a combination of perfect conductor and periodic boundary conditions, are studied. The results show that, under certain conditions, bi-periodic homogeneous cylindrical structures share some important properties with closed waveguides, namely, they can be described by means of a complete set of TEM, TE and TM modes. Considering the aforementioned properties, a mode-matching approach is proposed for the analysis of multilayered, bi-periodic structures with arbitrary metallization and dielectric thicknesses, as an accurate, efficient and versatile method. The application of the proposed approach is exemplified through the analysis of the incidence of plane waves at arbitrary angles on bi-periodic surfaces, the analysis and measurement of a waveguide simulator, and the characterization of a parallel-plate waveguide with a bi-periodically textured surface. Numerical results and measurements confirm the accuracy of this approach.