Magnetic Resonance Electrical Impedance Tomography (MREIT) is a late medical imaging modality visualizing static conductivity images of electrically conducting subjects. When we inject current into the object, it produces internal distributions of current density J and magnetic flux density B=(B _{x} ,B _{y} ,B _{z} ). By using an MRI scanner, we can measure B_{z} data where is the direction of the main magnetic field of the scanner. Conductivity images are reconstructed based on the relation between the injection current and B_{z} data. Recently, MREIT has rapidly progressed in its theory, algorithm, and experiment technique and now reached to the stage of in vivo animal experiments. In this talk, we present the basic concept of MREIT, a recent MREIT algorithm called localharmonic B_{z}-algorithm, and a software named CoReHA. Furthermore, we discuss problems in the area of computational mathematics and image processing which arise in MREIT.