TROJAN asteroids, which may outnumber the asteroids in the asteriod belt, are objects that orbit the Sun with the same mean semi-major axis as Jupiter, but lead or trail the position of Jupiter in its orbit by ~60°. One very interesting aspect of the Trojan swarms is that a significant number of asteroids are on orbits that analytic theory suggests should be unstable1. Here we present the results of long-term dynamical integrations of the Trojan asteroids, that enable us to investigate the stability of the swarm population. We find that the orbits of the swarm asteroids are not stable indefinitely-the gravitational effects of the giant planets have reduced the swarms' outer boundaries over time. We estimate that there are over 200 escaped Trojan asteroids with diameters >lkm currently roaming the Solar System, a few of which may be on Earth-crossing orbits.