In a pilot, randomized clinical trial, 24 stroke patients with moderate/severe impairments will receive non-invasive brain stimulation (repetitive Transcranial Magnetic Stimulation or rTMS) to one of two targets in the brain in conjunction with upper limb training for 2 days a week for 6 weeks. The primary outcome will be upper limb motor impairment, and secondary outcomes will be tests of functional abilities, proximal motor control, and patient-reported disability. Associated neural mechanisms will also be studied using neurophysiological and functional connectivity MRI techniques. Damage to ipsilesional corticospinal pathways will be indexed with diffusion tensor imaging (DTI).