The differential steering drive includes a drive differential interconnecting the drive shafts for the tracks and a steering differential for superimposing additive and subtractive rotations to the tracks for steering and pivot turning. In the preferred embodiment for high speed tracked vehicles, the drive differential is an all-gear no-clutch type limited-slip differential, and the steering differential is an unlimited-slip differential. The differentials are arranged to provide no-slip track operation traveling in straight paths or turning if at least one track has traction. In another embodiment for pivot-turning slow-moving off-road vehicles, both the drive and the steering differential are all-gear no-clutch type limited-slip differentials. Further, both embodiments preferably include additional left- and right-side all-gear no-clutch type limited-slip differentials for dividing the torque delivered to a respective pair of drive axles associated with each track.