Abstract: A high deflection, constant speed, universal joint has two end members which can be mounted on the adjacent ends of angularly-displaced shafts, has a first torque-transmitting support which is pivoted to one of those end members and which is directed transversely of the axis of that end member and which can oscillate relative to that end member about a first axis, has a second torque-transmitting support which is pivoted to the other of those end members and which is directed transversely of the axis of that other end member and which can oscillate relative to that other end member about a second axis, has variable-length torque-transmitting linkages which are pivotally secured to the first support for oscillation about a third axis and which are pivotally secured to the second support for oscillation about a fourth axis, and has actual or virtual means to cause the axes of the shafts to intersect and to cause the intersection of those axes to be equidistant from the first and second axes.