About 4 billion years ago, the terrestrial planets were bombarded by asteroids following an orbital shake-up of the outer Solar System. Lunar samples, planetary cratering records and dynamical models piece together an increasingly coherent view of this bombardment interval. During the first billion years of Solar System evolution, following planetary accretion, the rate of impact cratering was substantially higher than over the past 3.5 Gyr. However, the causes, magnitude and evolution of the early impact flux remain unknown. In particular, uncertainty persists about whether the largest impact basins on the Moon and the other terrestrial planets formed from a cataclysmic bombardment in a narrow window of time about 3.9 Gyr ago, as initially suggested by the lunar sample collection, or over a more extended period. Recent observations relating to this so-called Late Heavy Bombardment imply that the window of bombardment was not as narrow and intense as originally envisaged. Nevertheless, numerical simulations suggest that the rocky bodies left behind after planetary accretion are insufficient in number to form the youngest large impact basins 4.0 to 3.7 Gyr ago. One viable hypothesis for the formation of these basins is the delivery of impactors to the inner Solar System following the migration of the giant planets, but this scenario also faces challenges. Clarifying the magnitude and length of the Late Heavy Bombardment has implications across the full range of planetary geosciences, from understanding the dynamical evolution of the Solar System to surface conditions on the terrestrial planets early in their history.