In order to develop a lightweight and wearable bionic finger rehabilitation robot, a soft flexible finger rehabilitation robot actuated by shape memory alloy (SMA) wire was presented based on the analysis of the biological muscle structure and the actuation mechanism of fingers, and the kinematic and mechanical model were established. Taking gloves as the prototype of the structural design, the contraction of the muscle and tendon of fingers is simulated by controlling the contraction of the SMA wires, so as to assist the flexion and extension of fingers. The movement performance and grasping performance of the finger rehabilitation robot were studied. The experimental results show that the maximum flexion angle of the finger flexible rehabilitation robot is close to the normal hand joint angle, and the maximum fingertip force can reach 18N, which can complete the daily flexion/extension and grasping function.