The invention discloses a ball worm gear pair. Steel ball rolling in a spiral groove in the outer edge of a worm and worm gear is used for replacing meshed friction movement between threaded teeth of a worm and tooth faces of a worm gear to push the worm gear to rotate, and therefore the transmission efficiency and all technological performance indexes are greatly improved. The spiral groove in the worm and worm gear can be in the shape of a single arc or double arcs or a straight outline groove by means of the normal section of the worm and worm gear or the axial section of the worm. The enveloping manner is mainly a double enveloping manner and can also be a single enveloping manner. The steel ball circulating manner includes worm inner circulation and worm gear inner circulation. A steel ball rolls in a guide piece, the spiral groove, a reverse backflow device, an inner rolling way and an outer protection cover of the worm and worm gear in a circulating mode. The ball worm gear pair is used for transmitting movement and torque between two staggered shafts in space, the speed of movement can be reduced and can also be improved, the speed reduction movement with the crossed angle sigma of the shafts being 90 degrees is mostly commonly used, and the ball worm gear pair is widely applied in mechanical transmission.