A 26-Myr periodicity has recently been seen in the fossil record of extinction in the geological past1. At least two of these extinctions are known to be associated with the impact on the Earth of a comet or asteroid with a diameter of a few kilometres (refs 2, 3). We propose that the periodic events are triggered by an unseen companion to the Sun, travelling in a moderately eccentric orbit, which at its closest approach (perihelion) passes through the `Oort cloud' of comets which surrounds the Sun (ref. 4; see ref. 5 for a review and ref. 6 for a more recent analysis). During each passage this unseen solar companion perturbs the orbits of these comets, sending a large number of them (over 1 × 109) into paths which reach the inner Solar System. Several of these hit the Earth, on average, in the following million years. At present the unseen companion should be approximately at its maximum distance from the Sun, ~2.4 light yr, and it will present no danger to the Earth until approximately AD 15,000,000.