A vibration threading lathe has a cutting tool vibrated by an ultrasonic vibrator and supported by a tiltable frame which is so tiltable that the cutting tool can be inclined at the lead angle of the screw thread. As a result, screw cutting can be accomplished with remarkably low resistance to cutting, without burrs, and with a remarkably low rate of abrasive wear of the cutting tip. This lathe is highly suitable for threading articles of difficult-to-machine materials such as stainless steel, titanium, and beryllium copper, and particularly tubular articles of small diameters and having thin walls.