For Giuseppe Piazzi, the search paid off on the night of January 1, 1801. He started off the 19th century by discovering the asteroid Ceres. After the discovery of Ceres, many other asteroids were found. The discoverer of an asteroid has the privilege of naming it, but not until its orbit has been well-determined. Since the list of mythological names has long since been exhausted, anything goes as far as names are concerned.
Some asteroids lie beyond Jupiter's orbit, and some lie inside Mars's orbit.
Although all the big asteroids have been discovered and named, there are still many smaller asteroids that have not yet been discovered. One estimate is that the asteroid belt contains about 100,000 asteroids that are larger than 1 kilometer across. A typical asteroid is battered and irregular, looking remarkably like a potato that's been painted gray. The image below shows a pair of asteroids. The distance between asteroids is very much larger than the diameter of an asteroid.