Over the duration of the Kepler mission, KIC 8462852 was observed to undergo irregularly shaped, aperiodic dips in flux of up to ~20 per cent. The dipping activity can last for between 5 and 80 d.We characterize the object with high-resolution spectroscopy, spectral energy distribution fitting, radial velocity measurements, high-resolution imaging, and Fourier analyses of the Kepler light curve. We determine that KIC 8462852 is a typical main-sequence F3 V star that exhibits no significant IR excess, and has no very close interacting companions. In this paper, we describe various scenarios to explain the dipping events observed in the Kepler light curve.We confirm that the dipping signals in the data are not caused by any instrumental or data processing artefact, and thus are astrophysical in origin. We construct scenario-independent constraints on the size and location of a body in the system that are needed to reproduce the observations. We deliberate over several assorted stellar and circumstellar astrophysical scenarios, most of which have problems explaining the data in hand. By considering the observational constraints on dust clumps in orbit around a normal main-sequence star, we conclude that the scenario most consistent with the data in hand is the passage of a family of exocomet or planetesimal fragments, all of which are associated with a single previous break-up event, possibly caused by tidal disruption or thermal processing. The minimum total mass associated with these fragments likely exceeds 10-6 M⊕, corresponding to an original rocky body of >100 km in diameter. We discuss the necessity of future observations to help interpret the system.