Magnetic core properties, core loss, permeability, and saturation magnetic induction of bcc-nanocrystalline Fe-M-B (M=Zr, Hf, and Nb) alloys produced by annealing a melt-spun amorphous phase were investigated in a ring-shaped form with the aim of clarifying the application potential as a core material. The bcc alloys exhibit high saturation induction (Bs) from 1.49 to 1.63 T combined with high permeability (μe) from 22 000 to 32 000 at 1 kHz and 0.4 A/m. The bcc Fe-M-B (M=Zr, Hf, or Nb) alloys also show low core losses (W) from 1.4×10-1 to 2.1×10-1 W/kg at 50 Hz and 1.4 T and from 1.70 to 2.50 W/kg at 1 kHz and 1.0 T. The W values attained for the bcc Fe-M-B (M=Zr, Hf, and Nb) alloys are smaller by 60%- 90% at 50 Hz and 1.4 T and 50%-70% at 1 kHz and 1.0 T, as compared with those for an amorphous Fe78Si9B13 alloy in practical use as a transformer core material. The low W values for the bcc-nanocrystalline alloys are presumably due to the small anomaly factor comparable to a Co-based amorphous alloy. The comparison of the present data with those for the amorphous Fe78Si9B13 alloy indicates that the bcc-nanocrystalline Fe-M-B (M=Zr, Hf, and Nb) alloys are promising for practical use as core materials.