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I was able to obtain all of the gems while playing Diablo III.

Gem effects are stackable. If you place 3 gems of the same type in a sword, you will receive the bonus of all three. The first gem that is placed will determine the color of the weapon.

They can only be obtained by killing monsters, opening chests or using a Gem Shrine with no other gem in your inventory. Vendors do not normally sell gems, though if another player in a multiplayer game sells a gem to a vendor, you may be able to buy it. Vendors also will not typically pay you full price for socketed items.

Once inserted, gems cannot be removed from a socket, although they can be destroyed via a Horadric Cube recipe. The skulls of some demonic creatures, inscribed with eldritch symbols, are said to work much like gems. They were introduced to showcase the socketing system, which was expanded in the Lord of Destruction with the introduction of Runes and Jewels.

You can upgrade gems by using a Gem Shrine, with a gem in your inventory, or by using the Horadric Cube. You can use the Horadric Cube to convert gems into higher quality gems. Place 3 gems of one type into the Horadric Cube and click the Transmute button.

There are six types of gems, plus skulls, which are considered by the game to be gems for all purposes:

Amethyst

Diamond

Emerald

Ruby

Sapphire

Topaz

Skull

They come in five levels of increasing quality, which also determines their sell price to vendors:

Chipped (250)

Flawed (750)

Normal (2,500)

Flawless (5,000)

Perfect (15,000)