Stroke rehabilitation with the assistance of exoskeleton robots has the potential to facilitate the recovery of motor skills by guiding, supporting, and encouraging movement in the affected limbs. In this work, an upper-extremity exoskeleton actuated by the cable transmission mechanism, was developed for rehabilitation training of stroke patients. The exoskeleton is designed with three DOFs, including shoulder abduction/adduction, shoulder flexion/extension, and elbow flexion/extension. The main body of exoskeleton is comprised of three links and two cuffs, with a total weight of approximately 1.7 kg. Based on the mechanical design and cable arrangement, the kinematics and dynamics of the actuation cables were analyzed. In addition, a PID-based position control strategy and an impedance control strategy were designed based on the cable kinematics and dynamics. To investigate the application performance in passive motion assistance, a prototype of the cable-driven exoskeleton system was established, and the corresponding experiments were conducted with five subjects. In position control experiments, the measured joint trajectories exhibit similar variation trends and close peek angles to the demonstration trajectory, which is the main emphasis in rehabilitation scenarios. Meanwhile, it can be inferred that the time lag is the primary reason for the tracking error between the expected and measured trajectories. As to the impedance control experiments, the statistical results indicate that the employment of lower impedance parameters tends to protect the human-robot interaction safety by sacrificing the tracking accuracy of joint position.