Modern applications demand challenging operation requirements from electrical machines. Variable-flux machine (VFM) concepts are found to offer new opportunities for improved machine design. This paper reviews the VFM technology. Three topologies, mainly the parallel hybrid VFMs, series hybrid VFMs, and variable-flux flux intensifying type VFMs are discussed in detail. The variable-flux flux Intensifying machines are found to be the state-of-the-art VFM technology available to-date. The paper also reviews the magnetization process in the VFMs and relates this with the B-H curve of the magnetic material. The conventional PM machine mathematical model is adapted to represent the operation of the VFMs. The different operational limitations of the VFMs are explained and the different modes of operation of the VFMs have been reviewed and compared. The magnetization state manipulation techniques are reviewed. The AC magnetization is found to be the more preferable approach to reduce weight and manufacturing complexity of the VFMs. The trapezoidal current pulse based approach is popular in AC magnetization state manipulation, however, new advanced forms of current vector trajectory control techniques offer comparatively better magnetic state manipulation and higher speed capability for remagnetization of the magnets in the VFMs. The paper also reviews the different magnetization state estimation techniques that are useful during control systems development for the VFMs. The magnetic material for the VFM design is reviewed and potential alternative magnetic material that can satisfy the requirements of the VFMs have been identified. Finally, the current status, challenges, and the potential of VFMs technology are discussed in the conclusion of the paper.INDEX TERMS Variable flux machines, wide speed range, permanent magnet machines, field weakening, magnetic material, magnetization, demagnetization, machine design.