This letter proposes a soft wearable robot with shape memory alloy (SMA)-based artificial muscle to assist with upper limb motion. The SMA-based artificial muscle was designed by embedding a bundle of SMA coil springs into a stretchable coolant vessel, which allows active cooling of the SMA bundles through coolant circulation. By attaching SMA-based artificial muscles to a soft wearable brace, a soft wearable robot was designed to assist elbow flexion and forearm supination/pronation. To support the activities of daily living (ADLs), the proposed robot can produce more than 3 Nm torque in the direction of elbow flexion. In addition, the proposed robot can increase the forearm supination/pronation ROM by an average of 15.7 degrees for impaired users. Finally, to evaluate the effect of the assistance provided by the proposed system, experiments and system usability scale (SUS) assessments were conducted on a group of patients with stroke and spinal cord injuries (SCI). The results indicate that the robot improves not only the range of motion (ROM) but also the torque produced by the users. In addition, from the results of the user experiment, we found certain design issues that require further development for better performance of the robot.