Permanent magnets (PM) are critical components for electric motors and power generators. Key properties of permanent magnets, especially coercivity and remanent magnetization, are strongly dependent on microstructure. Understanding metallurgical processing, phase stability and microstructural changes are essential for designing and improving permanent magnets. The widely used PM for the traction motor in electric vehicles and for the power generator in wind turbines contain rare earth elements Nd and Dy due to their high maximum energy product. Dy is used to sustain NdFeB's coercivity at higher temperature. Due to the high supply risk of rare earth elements (REE) such as Dy and Nd, these elements are listed as critical materials by the U.S. Department of Energy and other international institutes. Other than Dy, finer grain size is also found to have effect on sustaining coercivity at higher temperature. A proper control of phase stability and microstructures has direct impact on mitigating REE supply risk. Compared to rare earth PMs, non-rare earth (non-RE) PMs typically have lower maximum energy products, however, given their small supply risks and low cost, they are being intensively investigated for less-demanding applications. The general goal for the development of non-RE PMs is to fill in the gap between the most cost-effective but low performing hard ferrite magnet and the most expensive but high performing RE PMs. In the past five years great progress has been made toward improving the microstructure and physical properties of non-RE PMs. Several new candidate materials systems were investigated, and some have showed realistic potential for replacing RE PMs for some applications. In this article, we review the science and technology of various types of non-RE materials for PM applications. These materials systems include Mn based, high magnetocrystalline anisotropy alloys (MnBi and MnAl compounds), spinodally decomposing alloys (Alnico), high-coercivity tetrataenite L10 phase (FeNi and FeCo), and nitride/carbide systems (such as α" based, high saturation magnetization Fe16N2 type phase and Co2C/Co3C acicular particle phase). The current status, challenges, potentials as well as the future directions for these candidates non-RE magnet materials are discussed.