Abstract: A constant velocity universal joint wherein its outer race has a larger radius of curvature than its inner race at the transverse sectional surface of the ball rolling groove at the point of contact between the ball and the ball rolling groove. Concave and convex surface portions of a ball rolling groove have a characteristic such that when the inner and outer races have the same radius of curvature at the transverse sectional surface of the ball rolling groove, the contact stress of the ball rolling groove of the outer race becomes lower than that of the inner race. By varying the radius of curvature on the transverse sectional surface of the ball rolling groove of the outer race from that of the inner race to render the contact stresses of the two grooves equal to each other, it is possible to facilitate working of the grooves and yet to achieve the same degree of precision finishes as has hitherto been achieved.