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Magnetic resonance electrical impedance tomography (MREIT) has been lately developed for high-resolution imaging of an electrical conductivity distribution inside the human body. In MREIT, we can avoid the inherent ill-posedness of the conductivity image reconstruction problem in electrical impedance tomography (EIT) by combining it with the current-injection MRI technique. MREIT utilizes an MRI scanner to measure internal magnetic flux density data induced by externally injected currents and produces multi-slice conductivity images with almost the same spatial resolution of conventional MR images. After the original ideas on MREIT in early 1990s, there has been rapid progress in its theory, algorithm, and experimental techniques. This paper reviews MREIT from the basics to the most recent research outcomes. It is expected that MREIT will be able to produce high-resolution conductivity images of the human body within one year. Suggesting future research directions, possible applications in biomedicine, biology, chemistry, and material science will be discussed.