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We propose a MoM technique using generalized curved parametric quadrilaterals of arbitrary geometrical order. In the new technique, the geometrical surface elements in the model can be both electrically large, with higher-order basis functions for current modeling, and curved. With this, the large-domain current-modeling efficiency is preserved even for surfaces with pronounced curvature (such as a sphere or a cylinder). The geometrical orders of the quadrilateral elements are entirely independent from the current-approximation orders of the basis functions, and the two sets of parameters of the higher-order model can be adopted at will for the best overall performance of the method. Numerical examples showing the efficiency and accuracy of the new technique are a metallic spherical scatterer, analyzed using four different higher-order models, and an antenna system consisting of two wire monopoles attached to a metallic cylinder.