A rotor (101, 201, 301) for a permanent magnet electrical machine, the rotor comprising: - a ferromagnetic core structure (102, 202, 302) comprising cavities dividing the ferromagnetic core structure into pole regions, a yoke region, and bridges (105, 106) between the pole regions and the yoke region, and - permanent magnets (107, 108, 207, 307, 308, 331) located in the cavities and arranged to produce magnetic fluxes penetrating the pole regions in radial directions so as to form magnetic poles of the rotor, the direction of the magnetic flux penetrating each pole region defining a direct-axis of the corresponding magnetic pole and a symmetry line between direct-axes of two adjacent magnetic poles defining a quadrature-axis between these two adjacent magnetic poles, wherein the cavities have portions (109, 110, 209, 210, 309, 310) which extend towards the quadrature-axis and are free from the permanent magnets so as to form non-ferromagnetic regions for reducing leakage fluxes of the permanent magnets, and the portions (109, 110, 209, 210, 310) of the cavities have a shape curving towards a geometrical rotation axis (111, 211, 311) of the rotor and radial height of the portions (109, 110, 209, 210, 309, 310) of the cavities is greater than radial thickness of parts of the permanent magnets (107, 108, 207, 208, 307, 308, 331) abutting on the portions of the cavities, characterized in that each of surfaces (112, 113) of the ferromagnetic core structure defining the portions (109, 110, 209, 210, 309, 310) of the cavities and facing towards the non-ferromagnetic regions defines a section curve in a plane perpendicular to the geometrical rotation axis of the rotor so that: a) the section curve is curvilinear on a portion of the section curve from a first point of the section curve farthest from the geometrical rotation axis of the rotor to a second point of the section curve nearest to the geometrical rotation axis of the rotor, b) the radius of curvature of the section curve depends on distance to the geometrical rotation axis of the rotor so that, on the portion of the section curve from the first point of the section curve to the second point of the section curve, the radius of curvature is an ascending function of the distance, and c) the section curve is substantially tangentially directed at the first point of the section curve and meets the relevant permanent magnet (107, 108, 207, 208, 307, 308, 331) at the first point of the section curve. - a ferromagnetic core structure (102, 202, 302) comprising cavities dividing the ferromagnetic core structure into pole regions, a yoke region, and bridges (105, 106) between the pole regions and the yoke region, and - permanent magnets (107, 108, 207, 307, 308, 331) located in the cavities and arranged to produce magnetic fluxes penetrating the pole regions in radial directions so as to form magnetic poles of the rotor, the direction of the magnetic flux penetrating each pole region defining a direct-axis of the corresponding magnetic pole and a symmetry line between direct-axes of two adjacent magnetic poles defining a quadrature-axis between these two adjacent magnetic poles, a) the section curve is curvilinear on a portion of the section curve from a first point of the section curve farthest from the geometrical rotation axis of the rotor to a second point of the section curve nearest to the geometrical rotation axis of the rotor, b) the radius of curvature of the section curve depends on distance to the geometrical rotation axis of the rotor so that, on the portion of the section curve from the first point of the section curve to the second point of the section curve, the radius of curvature is an ascending function of the distance, and c) the section curve is substantially tangentially directed at the first point of the section curve and meets the relevant permanent magnet (107, 108, 207, 208, 307, 308, 331) at the first point of the section curve.