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We present methods for analyzing and designing cylindrical electromagnetic metasurfaces with non-circular cross sections based on conformal transformations. It can be difficult to treat surfaces with non-canonical geometries since they generally do not admit straightforward solutions to the Helmholtz wave equation subject to the appropriate boundary conditions. This leads to the reliance on full wave numerical techniques which are only suitable for the analysis, but not the synthesis, of these surfaces. We address this issue by employing conformal transformations to map the physical space into a computational space in which the surface coincides with a circular cylinder. The electromagnetic boundary conditions on the surface remain intact under the transformations due to their angle-preserving nature. However, they are much more easily enforced. As a result, analytical modal solutions for the scattered fields are readily obtainable, which facilitate closed-form analysis and synthesis equations for general non-circular cylindrical metasurfaces. One important utility enabled by the proposed framework is the efficient identification of electromagnetic field distributions that satisfy local power conservation. This leads to passive and lossless surface designs, which are highly desirable in practice as they do not require active and/or lossy components.