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 to a peripherally outermost point 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; 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 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 of the arcuate cutting edges (51d1, 51d2) to each arcuate cutting edge (51d1, 51d2) to a rotation axis, 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 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; the radial rake angle of said arcuate cutting edge (51d1; 51d2) having the maximum value between said tip end and said rotationally most projecting point 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 via said rotationally most projecting point. arcuate cutting edges (51d1, 51d2) each extending from a tip end to a peripherally outermost point 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; 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 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 of the arcuate cutting edges (51d1, 51d2) to each arcuate cutting edge (51d1, 51d2) to a rotation axis, 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 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; the radial rake angle of said arcuate cutting edge (51d1; 51d2) having the maximum value between said tip end and said rotationally most projecting point 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 via said rotationally most projecting point.