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The scattering by large anisotropic objects is computed using the hybrid finite element-boundary integral-multilevel fast multipole algorithm (FE-BI-MLFMA). The validity of FE-BI-MLFMA for objects with off-diagonally anisotropic material is convincingly verified by comparing numerical results with analytical values. The numerical performance of the FE-BI-MLFMA is in detail investigated for anisotropic objects. It is shown that the numerical performance of the FE-BI-MLFMA is much worse for off-diagonally anisotropic objects than that for isotropic and uniaxially anisotropic objects. A node-edge element is employed for improving the efficiency of the FE-BI-MLFMA for anisotropic objects. The scattering characteristics by different large anisotropic objects are analysed by the FE-BI-MLFMA. Numerical results demonstrate that the anisotropic materials, especially off-diagonally anisotropic materials, can significantly change the radar cross section (RCS) pattern by complicated mechanisms, which depend on the structure of objects.