Over the past decade, there has been marked progress in electromagnetic property and mechanical property imaging techniques in which cross-sectional image reconstructions inside the human body are pursued. These techniques also have wider applications as imaging methods in medicine, biotechnology, non-destructive testing and the monitoring of industrial processes, and in other areas. Magnetic Resonance Electrical Impedance Tomography (MREIT) is an MR-based conductivity imaging technique at low frequencies. In MREIT, we inject currents into an imaging object after RF pulses and measure induced magnetic flux density data from MR phase images. Quasistatic Maxwell equations provide a relation between the measured magnetic flux density and the conductivity from which we can reconstruct cross-sectional conductivity images. Magnetic Resonance Electrical Property Tomography (MREPT) aims to reconstruct images of both conductivity and permittivity distributions inside the human body at MR frequency. In contrast to MREIT, MREPT does not require current injection since it is based on standard RF field mapping techniques to measure active magnetic RF field components. These modalities belong to interdisciplinary research areas incorporating mathematical theories and analyses of electromagnetism, MR physics and imaging methods, inverse problems and image reconstruction algorithms, numerical analyses and experimental techniques.