Utilizing results obtained over the past quarter century mainly with Japanese X-ray astronomy satellites, a review is given to some aspects of neutron stars (NSs), with a particular emphasis on the magnetic fields (MFs) of mass-accreting NSs and magnetars. Measurements of electron cyclotron resonance features in binary X-ray pulsars, using the Ginga and Suzaku observatories, clarified that their surface MFs are concentrated in a narrow range of (1–7) × 10⁸ T. Extensive studies of magnetars with Suzaku reinforced their nature as neutron stars with truly strong MFs, and revealed several important clues to their formation, evolution, and physical states. Taking all these results into account, a discussion is made on the origin and evolution of these strong MFs. One possible scenario is that the MF of NSs is a manifestation of some fundamental physics, e.g., neutron spin alignment or chirality violation, and the MF makes transitions from strong to weak states.