  Fast Radio Bursts are millisecond-duration astronomical radio pulses of
unknown physical origin that appear to come from extragalactic distances.
Previous follow-up observations have failed to find additional bursts at the
same dispersion measures (i.e. integrated column density of free electrons
between source and telescope) and sky position as the original detections. The
apparent non-repeating nature of the fast radio bursts has led several authors
to hypothesise that they originate in cataclysmic astrophysical events. Here we
report the detection of ten additional bursts from the direction of FRB121102,
using the 305-m Arecibo telescope. These new bursts have dispersion measures
and sky positions consistent with the original burst. This unambiguously
identifies FRB121102 as repeating and demonstrates that its source survives the
energetic events that cause the bursts. Additionally, the bursts from FRB121102
show a wide range of spectral shapes that appear to be predominantly intrinsic
to the source and which vary on timescales of minutes or shorter. While there
may be multiple physical origins for the population of fast radio bursts, the
repeat bursts with high dispersion measure and variable spectra specifically
seen from FRB121102 support models that propose an origin in a young, highly
magnetised, extragalactic neutron star.
