A fixed type constant velocity universal joint, comprising: an outer joint member having an inner surface provided with track grooves; an inner joint member having an outer surface provided with a plurality of track grooves paired with the track grooves of the outer joint member; a plurality of balls for transmitting torque, the plurality of balls being interposed between the track grooves of the outer joint member and the plurality of track grooves of the inner joint member; and a cage for holding the plurality of balls, the cage being interposed between the inner surface of the outer joint member and the outer surface of the inner joint member, the fixed type constant velocity universal joint comprising a lubricating grease sealed therein, the fixed type constant velocity universal joint comprising a fixed type constant velocity universal joint of an undercut free type in which a bottom surface of each of the track grooves of the outer joint member and a bottom surface of each of the plurality of track grooves of the inner joint member each comprise a curved portion (32a, 35b) and a straight portion (32b, 35a), wherein a surface roughness of each of the plurality of balls is set to Ra 0.15 μm or less, wherein the center of curvature of each of the track grooves of the outer joint member and the center of curvature of corresponding one of the plurality of track grooves of the inner joint member are offset to each other with respect to a plane, which is orthogonal to a joint axial center line and runs through a joint center, to the opposite sides by equal distances, characterized in that a surface roughness of a counterpart surface (32, 35) on which corresponding one of the plurality of balls rolls is set to be higher than the surface roughness of each of the plurality of balls, and wherein an additive composition of the lubricating grease comprises a base oil, a diurea compound, molybdenum dithiocarbamate, zinc dialkyldithiophosphate, melamine cyanurate, molybdenum disulfide, and calcium salt of alkylaromatic sulfonic acid, and wherein a center of curvature of each of the track grooves of the outer joint member and a center of curvature of corresponding one of the plurality of track grooves of the inner joint member are offset with respect to the joint axial center line in a radial direction by equal distances with respect respectively to corresponding one of the track grooves of the outer joint member and corresponding one of the plurality of track grooves of the inner joint member. an outer joint member having an inner surface provided with track grooves; an inner joint member having an outer surface provided with a plurality of track grooves paired with the track grooves of the outer joint member; a plurality of balls for transmitting torque, the plurality of balls being interposed between the track grooves of the outer joint member and the plurality of track grooves of the inner joint member; and a cage for holding the plurality of balls, the cage being interposed between the inner surface of the outer joint member and the outer surface of the inner joint member, the fixed type constant velocity universal joint comprising a lubricating grease sealed therein, the fixed type constant velocity universal joint comprising a fixed type constant velocity universal joint of an undercut free type in which a bottom surface of each of the track grooves of the outer joint member and a bottom surface of each of the plurality of track grooves of the inner joint member each comprise a curved portion (32a, 35b) and a straight portion (32b, 35a), wherein a surface roughness of each of the plurality of balls is set to Ra 0.15 μm or less, wherein the center of curvature of each of the track grooves of the outer joint member and the center of curvature of corresponding one of the plurality of track grooves of the inner joint member are offset to each other with respect to a plane, which is orthogonal to a joint axial center line and runs through a joint center, to the opposite sides by equal distances, characterized in that a surface roughness of a counterpart surface (32, 35) on which corresponding one of the plurality of balls rolls is set to be higher than the surface roughness of each of the plurality of balls, and wherein an additive composition of the lubricating grease comprises a base oil, a diurea compound, molybdenum dithiocarbamate, zinc dialkyldithiophosphate, melamine cyanurate, molybdenum disulfide, and calcium salt of alkylaromatic sulfonic acid, and wherein a center of curvature of each of the track grooves of the outer joint member and a center of curvature of corresponding one of the plurality of track grooves of the inner joint member are offset with respect to the joint axial center line in a radial direction by equal distances with respect respectively to corresponding one of the track grooves of the outer joint member and corresponding one of the plurality of track grooves of the inner joint member.