Just beyond the main asteroid belt there exists a population of small objects orbiting the Sun at the same heliocentric distance of Jupiter. The combined action of Jovian and Solar gravity creates a stable dynamical haven for these objects near the so-called Lagrangian equilibrium points, in honour to the French mathematician Joseph Louis Lagrange who first demonstrated their existence and stability. These two locations are special: they allow particles to stay in their vicinity ad infinitum. L4 leads the position of Jupiter along its orbit by 60 degrees while L5 lags behind the planet by the same amount.
The fact that a point in space is stable against perturbation is not a guarantee that there is a physical object there in the first place. There was considerable excitement among the dynamical astronomy community when the first such object was discovered by Max Wolf in 1906, now known as minor planet (588) Achilles. Since then over 1500 such Trojans have been discovered. They are thought to be remnant planetesimals from the time of Jupiter's formation. Having remained in pristine condition for the aeons since, their study can teach us a lot about the nature of the material that and the other giant planets are made of.
Since the existence of such stable regions hinges on the combined gravity of two masses -- the Sun and a planet -- then every planet in the solar system has them. In reality, however, gravitational perturbations by other bodies act to destabilize these regions rapidly. Up to the 1990s, only Jupiter was thought to be massive enough to keep its corresponding Trojan regions stable over a period of time comparable to the age of the solar system. The situation changed dramatically in 1990 when the object now known as (5261) Eureka was discovered. Soon after its discovery it was realized that this asteroid is, in fact, a Mars Trojan. Although the issue of its long-term stability has not, as yet, been conclusively resolved it demonstrates, at the very least, that for each of the inner planets a family of such attendants is a possibility which warrants further investigation. This possibility is rapidly turning into a certainty with the discovery of two more stable Martian Trojans, 1998 VF31 and 1999 UJ7, as well as several more candidate objects whose status will be ascertained when their orbits become sufficiently well-known.
In 1997 Earth gained its own companion, asteroid (3753) discovered more than 10 years ago, in 1986. Investigation of its motion showed that it does not adhere to the then existing theoretical models like the Jovian and Martian Trojans did. Instead it appeared to execute a complicated trajectory with respect to the Earth. Although not a Trojan per se, the existence of this object and several more of its kind show a richness of dynamical structure which can be borne out of the three body problem.