To date, there is no effective therapy for spinal cord injury, and many patients could benefit dramatically from at least partial restoration of arm and hand function. Despite a substantial body of research investigating intraspinal microstimulation (ISMS) in frogs, rodents and cats, little is known about upper-limb responses to cervical stimulation in the primate. Here, we show for the first time that long trains of ISMS delivered to the macaque spinal cord can evoke functional arm and hand movements. Complex movements involving coordinated activation of multiple muscles could be elicited from a single electrode, while just two electrodes were required for independent control of reaching and grasping. We found that the motor responses to ISMS were described by a dual exponential model that depended only on stimulation history. We demonstrate that this model can be inverted to generate stimulus trains capable of eliciting arbitrary, graded motor responses, and could be used to restore volitional movements in a closed-loop brain-machine interface.