Other stable patterns are possible, too: another one is what's called a 3:2 resonance pattern, asteroids whose motion gets confined to a basically triangular shape by the combined pull of Jupiter and the Sun. This group (for Jupiter) is called the Hilda Family, and their route forms a triangle with its three points at the two Lagrange points and at the point on Jupiter's orbit directly opposite it from the Sun.
None of these orbits are perfectly stable, because each of these asteroids is subject to pulling from everything in the Solar System; as a result, an asteroid can shift from the Lagrange points to the Hilda family, and from the Hilda family to the Asteroid Belt (not shown), especially if it runs into something and changes its course.
The reason that Pluto was demoted from planet to dwarf planet is that we realized that these things are not only numerous, but some of them are quite big. Some things we formerly called asteroids are actually bigger than Pluto, so the naming started to seem a little silly. So our Solar System has, in decreasing order of size, four gas giant planets (Jupiter, Saturn, Neptune and Uranus); four rocky planets (Earth, Venus, Mars, and Mercury); five officially recognized dwarf planets (Eris, Pluto, Haumea, Makemake, and Ceres); and a tremendous number of asteroids. (We suspect that there are actually about 100 dwarf planets, but the job of classifying what's an asteroid and what's actually a planet is still in progress -- see the "dwarf planet" link below if you want to know the details)
Ceres orbits in the Asteroid Belt, about halfway between Mars and Jupiter, just inside the triangle of the Hilda Family; Pluto and Haumea are both in the distant Kuiper Belt, outside the orbit of Neptune but shepherded by its orbit in much the same way that the Hildas are shepherded by Jupiter; Makemake is what's called a "cubewano," living in the Kuiper Belt but unshepherded, orbiting independently; and Eris is part of the Scattered Disc, the even more distant objects whose orbits don't sit nicely in the plane of the Solar System at all, having been kicked out of that plane by (we believe) scattering off large bodies like Jupiter.
But mostly, I wanted to share this to show you how things orbit. This picture comes from the amazing archive at http://sajri.astronomy.cz/asteroidgroups/groups.htm, which has many other such pictures, and comes to me via +Max Rubenacker.
More information about all of these things: