A wrist driving mechanism capable of arranging a large amount of cables and pipes in a robot arm. A robot wrist mechanism having three degrees of freedom is driven by first, second and third drive shafts for transmitting driving forces of first, second and third motors M4, M5 and M6, respectively. The first drive shaft is a hollow shaft for transmitting a rotational driving force from the motor M4 to a first wrist element supported rotatably around the first axis B1. The second and third drive shafts are arranged eccentrically with the first axis B1 in an inner space of the first drive shaft. The second drive shaft transmits a rotational driving force from the motor M5 to a second wrist element supported rotatably around the second axis B2, and the third drive shaft transmits a rotational force from the motor M6 to a third wrist element supported rotatably around the third axis B3. A first gear and a third gear as planet gears and a second gear and a fourth gear as sun gears are provided for avoiding complication of structure due to the eccentric arrangement of the second and third drive shafts.