Multiple electromagnetic (EM) scattering by arrays of infinite cylinders is essential for understanding wave interactions in structured media, with significant applications in antenna design, active ferromagnetic devices, tunable light transport, plasmonics, metasurfaces, metamaterials, EM shielding, and other related areas. In this work, we treat the multiple scattering problem by cylindrical arrays with the method of auxiliary sources (MAS). We describe in detail the use of MAS for various cylindrical arrays, including perfectly electric conducting (PEC), dielectric, and core–shell PEC–dielectric rectangular schemes. We also discuss how the MAS can be applied for the respective circular configurations. Excitations by a current filament or a plane wave are both considered. To highlight the robustness of the method, we apply the MAS for a potential application of cylindrical arrays achieving EM shielding. We calculate the shielding effectiveness showing that rectangular or circular cylindrical arrays may be used to significantly reduce the penetrating fields and thus provide protection to sensitive electronic devices from EM interference. To establish the validity of the method, the results obtained from MAS are compared to those obtained from the HFSS commercial software. Collectively, the developed method and the findings of this work enhance both the theoretical and the practical understanding of multiple EM scattering, offering comprehensive insights applicable to various configurations and material properties.