Hardstone carving is the general term for the carving for artistic purposes of semi-precious stones such as Jade, Rock Crystal, agate, onyx, jasper, and for an object made in this way. Normally the objects are small and the category overlaps with both jewellery and sculpture.
From the Neolithic period until about the 19th century such objects were among the most highly prized in a wide variety of cultures. The types of objects carved have included those ritual or religious purposes, engraved gems as signet rings and other kinds of seal, handles, belt hooks and similar items, vessels and purely decorative objects.
The art is very ancient going back to the Indus Valley Civilisation (3300-1300 BCE) and beyond. Chinese Jade carving can be found from the Neolithic Lianzhu culture (3400 - 2250 BCE) onwards. From the early civilizations of the near East descended the carving of vessels and small statues in Ancient Greece, Ancient Rome, and subsequent western art. One of the most notable producers of hard stone carving was Faberge who made his reputation with small hardstone figures of animals and people, and small vases with a few flowers.
Here are some great examples of Hard Stone Carving that I have found: