A rotor for a permanent magnet electrical machine includes a ferromagnetic core structure having cavities which divide the ferromagnetic core structure into pole regions (103), a yoke region (104), and bridges (105, 106) between them. The rotor includes permanent magnets (107, 108) located in the cavities and arranged to produce magnetic fluxes penetrating the pole regions so as to form magnetic poles of the rotor. The cavities have portions which extend towards the quadrature-axis (Q) and are free from the permanent magnets. Thus, non-ferromagnetic regions for reducing leakage fluxes of the permanent magnets are being formed. Furthermore, the portions of the cavities have a shape curving towards the geometrical rotation axis of the rotor. As the portions of the cavities have the shape curving towards the rotation axis, the cavities can be made longer and the routes of the leakage.