The invention provides a vertical tuned mass ball screw type inerter eddy current damper. The vertical tuned mass ball screw type inerter eddy current damper comprises an outer frame, a vertically-arranged guide rod, a support frame, a mass block, a spring, a permanent magnet, a conductor plate and an iron plate, and a set gap is reserved between the permanent magnet and the conductor plate; and the damper further comprises a vertically-arranged ball screw, a nut, a flywheel and a thrust bearing. When vibration energy of the controlled structure is transmitted to the damper, the mass block iscaused to vibrate up and down, the flywheel is driven to rotate at a high speed through the ball screw, the conductor plate cuts magnetic induction lines of the permanent magnet, and the eddy currentdamping effect is formed; and the back iron can reduce magnetic flux leakage and improve the damping energy consumption efficiency. According to the vertical tuned mass ball screw type inerter eddy current damper, a gap between the permanent magnet and the conductor plate remains unchanged in the moving process of the mass block, so that the distance between the permanent magnet and the conductorplate can be set to be very small, and very large damping force can be generated; the mass block does linear motion, the damping force has no torque on the mass block, the pressure between the mass block and the guide rod is not increased, and the motion is smooth.