A ball-end mill comprising, in a tip end portion of an end mill body : arcuate cutting edges (51d1, 51d2) each extending from a tip end (P) to a peripherally outermost point (S) along a curve in an S shape when viewed from the front side; spirally-shaped, peripheral cutting edges (51k1, 51k2) each smoothly connected to each arcuate cutting edge (51d1, 51d2); and a convex rake face (51e1, 51e2) of each arcuate cutting edge (51d1, 51d2) protruding forward in a rotation direction (R); each of said arcuate cutting edges (51d1, 51d2) having a radial rake angle, being an angle of each rake face (51e1, 51e2) to a straight line radially extending from the center point (O) of the arcuate cutting edges (51d1, 51d2) to each arcuate cutting edge (51d1, 51d2), characterised by said angle meeting the condition of β < α ≤ γ, wherein α is a radial rake angle at a radial angle, being an angle of the straight line radially extending from the center point (O) of the arcuate cutting edges (51d1, 51d2) to each arcuate cutting edge (51d1, 51d2) to a rotation axis (L), of 5°, β is a radial rake angle at a radial angle of 90°, and γ is a radial rake angle at a rotationally most projecting point (Q) of said arcuate cutting edge (51d1, 51d2), being the point, at which each arcuate cutting edge (51d1, 51d2) is most projecting in the rotation direction (R); the radial rake angle of said arcuate cutting edge (51d1; 51d2) having the maximum value between said tip end (P) and said rotationally most projecting point (Q) and in a radial angle range of 12-40°; and said radial rake angle changing along such a smooth curve that said radial rake angle gets gradually smaller in a range from a point of said maximum value to said peripherally outermost point (S) via said rotationally most projecting point (Q). arcuate cutting edges (51d1, 51d2) each extending from a tip end (P) to a peripherally outermost point (S) along a curve in an S shape when viewed from the front side; spirally-shaped, peripheral cutting edges (51k1, 51k2) each smoothly connected to each arcuate cutting edge (51d1, 51d2); and a convex rake face (51e1, 51e2) of each arcuate cutting edge (51d1, 51d2) protruding forward in a rotation direction (R); each of said arcuate cutting edges (51d1, 51d2) having a radial rake angle, being an angle of each rake face (51e1, 51e2) to a straight line radially extending from the center point (O) of the arcuate cutting edges (51d1, 51d2) to each arcuate cutting edge (51d1, 51d2), characterised by said angle meeting the condition of β < α ≤ γ, wherein α is a radial rake angle at a radial angle, being an angle of the straight line radially extending from the center point (O) of the arcuate cutting edges (51d1, 51d2) to each arcuate cutting edge (51d1, 51d2) to a rotation axis (L), of 5°, β is a radial rake angle at a radial angle of 90°, and γ is a radial rake angle at a rotationally most projecting point (Q) of said arcuate cutting edge (51d1, 51d2), being the point, at which each arcuate cutting edge (51d1, 51d2) is most projecting in the rotation direction (R); the radial rake angle of said arcuate cutting edge (51d1; 51d2) having the maximum value between said tip end (P) and said rotationally most projecting point (Q) and in a radial angle range of 12-40°; and said radial rake angle changing along such a smooth curve that said radial rake angle gets gradually smaller in a range from a point of said maximum value to said peripherally outermost point (S) via said rotationally most projecting point (Q). <EOS>