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Systems of singular integral-integrodifferential equations are studied that pertain to 2-D oblique scattering and to hybrid-wave propagation in presence of a dielectric cylinder with arbitrarily shaped smooth boundary. These systems, having the tangential to the surface of the cylinder components of the electric and magnetic fields as the unknowns, are solved via fast, highly accurate algorithms that rely on the Nystrom method (NM). Because of our specialized treatment, the present solution technique has the following characteristics: (1) it fully accounts for the singular nature of the kernels, (2) it appears to converge exponentially, and (3) it yields simple closed-form expressions for all matrix elements. In addition, an extension of the analysis is presented to account for composite dielectric cylinders that contain arbitrarily shaped dielectric or conducting cylindrical inclusions. Numerical examples and case studies illustrate the simplicity, flexibility, and efficiency of the algorithms. Exhaustive comparisons with available results for several special cases serve to test the correctness of the implementation and bring to light the extremely high accuracy of our algorithms.