Recent years have seen the discovery of several objects in stable orbits in the outer Solar System; these bodies include objects in the Kuiper belt (also known as the Kuiper-Edgeworth belt) as well as the Centaurs. Moreover, another region of orbital stability has been identified between the orbits of Uranus and Neptune. Here we report evidence from numerical simulations of zones of orbital stability in the inner Solar System. We find that there are two possible long-lived belts of asteroids. The first region lies between the Sun and Mercury, in the range 0.09-0.21 astronomical units, where remnant planetesimals may survive for the age of the Solar System provided that their radii are greater than ~0.1 kilometres. The second region of stability is between Earth and Mars (range 1.08-1.28 astronomical units), where a population of bodies that are on circular orbits may survive. A search through the catalogues of near-Earth objects reveals an excess of asteroids with low eccentricities and inclinations occupying this latter region: several examples are the recently discovered objects 1996 XB27, 1998 HG49 and 1998 KG3.