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Previous work on the subject of electromagnetic scattering has shown that the extended Born approximation (EBA) is more accurate than the first-order Born approximation with approximately the same operation count. However, the accuracy of the EBA degrades in cases when the source is very close to the scatterer, or when the electric field exhibits significant spatial variations within the scatterer. This paper introduces a generalized extended Born approximation (GEBA) and its high-order variants (Ho-GEBA) to efficiently and accurately simulate electromagnetic scattering problems. We make use of a generalized series expansion of the internal electric field to construct high-order terms of the generalized extended Born approximation (Ho-GEBA). A salient feature of the Ho-GEBA is its enhanced accuracy over the Born approximation and the EBA, even when only the first-order term of the series expansion is considered in the approximation. This behavior is not conditioned by either the source location or the spatial distribution of the internal electric field. A unique feature of the Ho-GEBA is that it can be used to simulate electromagnetic scattering due to electrically anisotropic media. Such a feature is not possible with approximations of the internal electric field that are based on the behavior of the background electric field. Three-dimensional (3-D) models of electromagnetic scattering are used to benchmark the efficiency and accuracy of the Ho-GEBA, including comparisons against the first-order Born approximation and the EBA.