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The controlled motion of a series of domain walls along magnetic nanowires using spin-polarized current pulses is the essential ingredient of the proposed magnetic racetrack memory, a new class of potential non-volatile storage-class memories. Using permalloy nanowires, we achieved the successive creation, motion, and detection of domain walls by using sequences of properly timed, nanosecond-long, spin-polarized current pulses. The cycle time for the writing and shifting of the domain walls was a few tens of nanoseconds. Our results illustrate the basic concept of a magnetic shift register that relies on the phenomenon of spin-momentum transfer to move series of closely spaced domain walls.