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The Darkmoon Faire occurs once a month (starting the first Sunday of the month and lasting until the next Saturday).

There are numerous daily quests, in the form of silly games, that you can complete on the Faire grounds for Darkmoon Prize Tickets.  These tickets can be used to purchase a number of things including pets, mounts, heirlooms and transmog gear.

In addition to the daily games, there are monthly quests that will grant you +5 skillups to all primary and secondary professions.  Some of them require additional reagents that you will have to purchase ahead of time:

For Leatherworking bring: x5 Coarse thread, x5 Blue Dye, x10 Shiny Bauble
For Tailoring bring: x1 each of Coarse thread, Red Dye, Blue Dye
For Alchemy bring: x5 Moonberry Juice
For Inscription bring: x5 Light Parchments
For Archaeology, have x15 Fossil fragments gathered.
For Cooking, bring: x5 Simple Flour

Blacksmithing, Engineering, Enchanting, Jewelcrafting, First aid and Fishing do not require any additional materials to complete, nor do the gathering professions Mining, Herbalism, and Skinning.

March 2015 added:

•  Darkmoon Race Enthusiast
•  Darkmoon Racer Novice
•  Darkmoon Racer Jockey
•  Darkmoon Racer Leadfoot
•  Darkmoon Racer Roadhog (awards  Fire-Eater's Vial)

•  Ace Tonk Commander (awards  Darkmoon Tonk Controller)

•  Triumphant Turtle Tossing (awards  Darkmoon Ring-Flinger)

In April, new achievements, a new toy, a new battle pet, and a limited item are available.  (Don't worry, you'll still be able to obtain everything from March; just choose the Racing Strider instead of the new Rocketeer racer.)

•  Blast Off!
•  Rocketeer: Bronze
•  Rocketeer: Silver
•  Rocketeer: Gold (awards  Everlasting Darkmoon Firework)

•  Rocket Man
•  Big Rocketeer: Bronze
•  Big Rocketeer: Silver
•  Big Rocketeer: Gold (awards  Race MiniZep Controller)

•  Big Race Enthusiast
•  Big Race Novice
•  Big Race Jockey
•  Big Race Leadfoot
•  Big Race Roadhog (awards  Checkered Flag)

May will also have new achievements and a new toy.  (You will be able to complete the stuff from March and April still, just choose the appropriate racer.)

•  Wayfarer
•  Wanderluster: Bronze
•  Wanderluster: Silver
•  Wanderluster: Gold (awards  Attraction Sign)

For June (same from previous applies):

•  Go-Getter
•  Powermonger: Bronze
•  Powermonger: Silver
•  Powermonger: Gold (awards  Gaze of the Darkmoon)

Don't forget that with the 6.1 update, heirlooms offered by Daenrand Dawncrest now cost less tickets!  If you prefer not to spend gold at Estelle Gendry / Krom Stoutarm, this is an alternative source.  Keep in mind when purchasing, however, that there is a finite amount of   Darkmoon Prize Ticket that can be earned each month (per character).

The following were added to the event in Patch 6.0.2:

Dailies

• Firebird's Challenge (includes achievement reward toy  Blazing Wings)
• A New Darkmoon Challenger! (chance to obtain  Syd the Squid)
  

Items

• These are all sold by Chester

 Last Deck of Nemelex Xobeh
 Haunting Memento
 Nobleman's Coat
 Nobleman's Pantaloons
 Noble's Fancy Boots
 Noblewoman's Finery
 Noblewoman's Skirt

NPC

• Erinys (drops level 100 quest-starter item which rewards  Ring of Broken Promises)