Fast Radio Bursts are millisecond-duration astronomical radio pulses of unknown physical origin that appear to come from extragalactic distances [1][2][3][4][5][6][7][8] . 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 9 . The apparent non-repeating nature of the fast radio bursts has led several authors to hypothesise that they originate in cataclysmic astrophysical events 10 . Here we report the detection of ten additional bursts from the direction of FRB 121102, using the 305-m Arecibo telescope. These new bursts have dispersion measures and sky positions consistent with the original burst 4 . This unambiguously identifies FRB 121102 as repeating and demonstrates that its source survives the energetic events that cause the bursts. Additionally, the bursts from FRB 121102 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 FRB 121102 support models that propose an origin in a young, highly magnetised, extragalactic neutron star 11,12 .2 FRB 121102 was discovered 4 in the PALFA survey, a deep search of the Galactic plane at 1.4 GHz for radio pulsars and fast radio bursts (FRBs) using the 305-m William E. Gordon Telescope at the Arecibo Observatory and the 7-beam Arecibo L-band Feed Array (ALFA) 13,14 . The observed dispersion measure (DM) of the burst is roughly three times the maximum value expected along this line of sight in the NE2001 model 15 of Galactic electron density, i.e. β DM ≡ DM FRB /DM Gal Max ∼ 3, suggesting an extragalactic origin.Initial Arecibo follow-up observations were limited in both dwell time and sky coverage and resulted in no detection of additional bursts 4 . In 2015 May and June we carried out more extensive follow-up using Arecibo, covering a ∼ 9 radius with a grid of six ALFA pointings around the then-best sky position of FRB 121102 (Figure 1 and Extended Data Table 1 and 2). As described in the Methods, high-time-resolution, total intensity spectra were recorded, and the data were processed using standard radio-frequency interference (RFI) excision, dispersion removal, and single-pulse-search algorithms implemented in the PRESTO software suite and associated data reduction pipelines 14,16,17 .We detected 10 additional bursts from FRB 121102 in these observations. The burst properties, and those of the initial FRB 121102 burst, are listed in Table 1. The burst intensities are shown in Figure 2. No other periodic or single-pulse signals of a plausible astrophysical origin were detected at any other DM. Until the source's physical nature is clear, we continue to refer to it as FRB 121102 and label each burst chronologically starting with the o...