Abstract: Apparatus for twisting a travelling strand includes a rotary structure rotatable about a rotational axis X--X and a series of pulleys mounted on the structure in an array disposed about the rotational axis X--X with each pulley in the series mounted on the structure for rotation about an individual axis of pulley rotation which is generally parallel to the axis X--X and moves around that axis on rotation of the structure. An input strand guide pulley guides an input strand to be twisted onto a path in which it will successively engage the pulleys and a strand exit guide pulley guides the strand from that path into an exit path along the axis X--X. The rotary structure is driven through a drive shaft and the pulleys are rotated by gearing.