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In this paper we give an overview of a recently developed method for solving an inverse Maxwell problem in environmental and geophysical imaging. Our main focus is on low-frequency cross-borehole electromagnetic induction tomography (EMIT), although related problems arise also in other applications in nondestructive testing and medical imaging. In typical applications (e.g. in environmental remediation), the isotropic or anisotropic conductivity distribution in the earth needs to be reconstructed from surface-to-borehole electromagnetic data. Our method uses a backpropagation strategy (based on adjoint fields) for solving this inverse problem. The method works iteratively, and can be considered as a nonlinear generalization of the Algebraic Reconstruction Technique (ART) in X-ray tomography, or as a nonlinear Kaczmarz-type approach. We will also propose a new regularization scheme for this method which is based on a proper choice of the function spaces for the inversion. A detailed sensitivity analysis for this problem is given, and a set of numerically calculated sensitivity functions for homogeneous isotropic media is presented.