Internal dimensions for fixed-type constant velocity universal joints are defined to optimize the geometry of the components of the joints and to readily achieve a larger operating angle of the joints. In the joint, the track grooves of an outer ring and an inner ring are tapered by an angle not exceeding 12°. The ratio (f/PCR) of the cage offset amount f to PCR is greater than 0 and smaller than or equal to 0.12. The ratio (Do/d) of the cage outer diameter (Do) to the ball diameter (d), the ratio (t/d) of the cage wall thickness (t) to the ball diameter (d), and the ratio (w/d) of the cage width (w) to the ball diameter (d) are such that 3.9≦Do/d≦4.1, 0.31≦t/d≦0.34, and 1.8≦w/d≦2.0, respectively. The end of the spherical outer surface of the cage adjacent to the open end is axially extended, and the end of the spherical inner surface of the cage adjacent to the open end is tapered so that the tapered surface has an increasing diameter toward the end of the spherical outer surface adjacent to the open end. Part of the spherical outer surface of the cage is cut on the open end side of pockets, and the outer diameter of the cage across the two opposing planer cut faces is smaller than the spigot diameter of the outer ring.