Stones of Barenziah

 are gems in The Elder Scrolls V: Skyrim.

Description
Initially, each Stone of Barenziah is regarded as an Unusual Gem. Finding one of these stones initiates a Miscellaneous Quest that indicates that the Dragonborn must have the gem appraised before it can be used. Only a few people are able to appraise these Gems; two of which are Maul in Riften and Vex, part of the Thieves Guild, also in Riften.

Upon getting it appraised, the Dragonborn learns that a total of 24 Stones of Barenziah exist, and they are only valuable as a complete set. Most are found as part of extra jobs for the Thieves Guild. The reward for collecting all of these items is the perk, Prowler’s Profit.

Locations

 * NOTE: taking any stone that is in someone else's house or castle (in a city) is considered a crime. Becoming thane might effect whether it is or not.
 * Solitude, Proudspire Manor: In the master bedroom. Proudspire can be accessed by Pickpocketing the Solitude Fletchers Key from the Solitude Fletcher. Use this key to enter the Porch Door into Proudspire Manor. The Stone is found in the upstairs Master bedroom, on a chest of drawers, on left as you enter the room. (if you use the pick pocketed key method then when you do buy the house your weapon plaques and racks won't work due to a glitch. This glitch applies to all homes if you enter them before buying home and ALL decorations.) If this is not possible, the property must be bought by the Dragonborn.
 * Solitude, Blue Palace: On the bedside table in Jarl Elisif's quarters.
 * Reeking Cave Under a ledge, next to the body of a conjurer. Note: prior to patch 1.4, this stone used to be located in the Thalmor Embassy. If the player has already obtained the stone in the Thalmor Embassy during the main quest before installing patch 1.4, he will have to return to the reeking cave to fetch the stone as the first stone will have disappeared from the inventory. As of Feb. 8 2012 patch 1.4 is not yet live on XBOX or PS3. On these systems, this stone can currently only be reached after the completion of Diplomatic Immunity with the aid of one of several glitches, an example of which can be seen here.
 * Dainty Sload: On a table in the First Mate's Quarters (ship is docked between Solitude Lighthouse & Solitude).
 * College of Winterhold: On a shelf in Arch-Mage's Quarters, which becomes accessible during the Under Saarthal quest.
 * Yngvild: In the chamber behind the throne in throne room.
 * Hob's Fall Cave: In the sleeping area (near the alchemy lab).
 * Markarth, Treasury House: Next to bed in the master bedroom.
 * Markarth, Understone Keep: On a table in a locked room to the left in the Dwemer Museum (Taking this one is considered a crime).
 * Dead Crone Rock: On a makeshift altar in front of the Word Wall at the top of the final tower.
 * Whiterun, Jorrvaskr: In Kodlak's bedroom.
 * Whiterun, Hall of the Dead: Enter the Catacombs, make an immediate left, then an immediate right, down the ramp, first crypt on the left.
 * Whiterun, Dragonsreach: In the Jarl's bedroom. Removing the stone is not considered a crime. (Confirmed on Xbox 360 3/2/2012)
 * Rannveig's Fast: On a table near the watery prison.
 * Fellglow Keep: On a counter in the workroom/workshop at the top of the front foyer (next to the alchemy lab).
 * Windhelm, House of Clan Shatter-Shield: Upstairs, in the first bedroom on the left. Removing the Stone is not considered a crime. (Confirmed on Xbox 20/01/2012)
 * Windhelm, Palace of the Kings: On a table in Wuunferth the Unliving's quarters (take the first door on the left in the main hall leading upstairs, his room is at the end of the hall). Removing the Stone is considered a crime. (Confirmed on Xbox 27/01/2012)
 * Stony Creek Cave: In the Bandit Wizard's cavern.
 * Ansilvund: Near Fjori's ghost in the burial chambers.
 * Sunderstone Gorge: On the altar in front of the Word Wall. (word of power mission to find it calls the area Greywater Gorge)
 * Dark Brotherhood Sanctuary : On the dresser in Astrid's room. SPOILER: The stone can still be obtained after the Dark Brotherhood is destroyed.
 * Pinewatch: In Pinewatch, downstairs across from fireplace there is a button to the right of the shelves on the wall, press it, and the shelves swing open revealing a tunnel to the pinewatch bandit sanctuary. Once inside the bandit sanctuary you will reach a room with a bandit and a bunch of empty draugr sarcophagi. To the right of this room is a small room with a treasure chest inside. The stone of barenziah is on a shelf directly across from the treasure chest.
 * Riften, Mistveil Keep: In the Jarl's chambers on the end table next to her bed on left side.
 * Black Briar Lodge: In the upstairs master bedroom.

After handing in the 24 Stones of Barenziah, the Dragonborn is asked to recover the Crown of Barenziah from Tolvald's Cave. Vex reveals that the Crown of Barenziah is an item that enhances a thief's ability. By recovering the stones, Vex can restore the paragon to its full strength. Click here for the Detailed video guide or the Quick Guide to finding ALL the stones.

Reward
After you have completed No Stone Unturned, Vex rewards the Dragonborn with the Prowler's Profit. Afterward, the fully restored Crown of Barenziah sits on the mannequin in the Thieves Guild.

Behind the scenes

 * It is not necessary to buy Proudspire Manor. Instead, steal the Fletcher key(from they guy inside the fletcher shop, arrow shop in Solitude), then open the back door to the Manor. It isn't necessary to steal the key if you have the Wax Key perk, simply pick the lock on the Fletcher shop in solitude and the right key will automatically be added to your inventory.
 * For gems found inside dungeons where you might be fighting NPCs - they can be knocked off their table, shelf, pedestal or whatever by spells, arrows, bodies running into them, walking over them etc. and sometimes fall into inaccessible locations like a deep furrow in the ground or underneath furniture. In such a case, you may need to return to the area after a few in-game days and hope that the item locations are reset back to where they were at initially.

Bugs

 * Fellglow Keep: For some reason, the gem may not appear on the counter it is supposed to be on.
 * This bug happened to us, but we went back to Solitude in the Jarls room and another stone was there (letting us complete the 24 stones).
 * Calixto's House of Curiosities: If you have checked all 24 locations and are still missing a Stone, check Calixto's museum. Sometimes one can be found on the shelves here.
 * Entering the Dwemer Museum after having completed Hard Answers will start that quest again. To move the stone to your location without entering the museum, you can use the console command prid 7F909 followed by moveto player.
 * Alternatively, you can complete the Hard Answers quest again to score another Nightingale Blade. Subsequent Thieves Guild quests will not start again. Not tested, but this is potentially a way to obtain unlimited Nightingale Blades. (You can't complete Darkness Returns if you did Hard Answers twice.)
 * Before being appraised, stones will stack under the name "Unusual Gem." After appraisal they will take the name "Stone of Barenziah" and will no longer stack.
 * Have the gems appraised before finding them all. Otherwise, it may not be possible to complete the quest.
 * Maul's dialogue may not show up, inhibiting the initiation of the quest.