Current-controlled magnetic domain wall motion has opened the possibility of a novel type of shift register memory device, which has been optimistically predicted to replace existing magnetic memories. Owing to this promising prospect, intensive work has been carried out during the last few decades. In this article, we first review the progress in the study of current-induced magnetic domain wall motion. Underlying mechanisms behind the domain wall motion, which have been discovered during last few decades, as well as technological achievements are presented. We then present our recent experimental results on the real-time detection of current-driven multiple magnetic domain wall motion, which directly demonstrates the operation of a magnetic domain wall shift register. (C) 2017 The Japan Society of Applied Physics