Paralysis of the upper extremity severely restricts independence and quality of life after spinal cord injury. Regaining control of hand and arm movements is the highest treatment priority for people with paralysis, 6-fold higher than restoring walking ability. Nevertheless, current approaches to improve upper extremity function typically do not restore independence. Spinal cord stimulation is an emerging neuromodulation strategy to restore motor function. Recent studies using surgically implanted electrodes demonstrate impressive improvements in voluntary control of standing and stepping. Here we show that transcutaneous electrical stimulation of the spinal cord leads to rapid and sustained recovery of hand and arm function, even after complete paralysis. Notably, the magnitude of these improvements matched or exceeded previously reported results from surgically implanted stimulation. Additionally, muscle spasticity was reduced and autonomic functions including heart rate, thermoregulation, and bladder function improved. Perhaps most striking is that all six participants maintained their gains for at least three to six months beyond stimulation, indicating functional recovery mediated by long-term neuroplasticity. Several participants resumed their hobbies that require fine motor control, such as playing the guitar and oil painting, for the first time in up to 12 years since their injuries. Our findings demonstrate that non-invasive transcutaneous electrical stimulation of the spinal networks restores movement and function of the hands and arm for people with both complete paralysis and long-term spinal cord injury.