Nether wart is a fairly difficult resource to obtain initially. They are only found in nether fortresses and usually in small supply. Once found however, they can be planted and grown on soul sand anywhere, in any dimension (Nether, Overworld or End).
Video
Tutorials/Nether wart farming/video
Locations For A Nether Wart Farm
Before 1.3.1 update nether wart could only grow in the Nether. In order to place a farm in the nether, safety must be your primary concern, avoiding ghasts to blow up your farm with its fireballs when they get you in sight. To secure your farm, build walls and a roof with blast-resistant materials, such as cobblestone and nether brick. Iron bars or nether brick fence will also be useful for windows.
As of the 1.3.1 update, nether wart can also be grown in the Overworld. Nether wart does not need light to grow up, but you have to be aware of creepers, so you should build your farm on a lit and enclosed place. To speed up growth the farm should be someplace where it's often in your chunk update radius.
Design
No fully automatic designs can be built, as nether wart must be planted manually. Only semi-automatic designs can be built, meaning that only the collection can be automated. On console, designs must take into account the 200 item entity limit. If more than 200 item entities are created simultaneously the game will delete them until only 200 remain.
Dispenser powered farms
In the following design a row of dispensers activated by redstone circuitry make water flow down to a collection area with hoppers, that stores the nether wart into chests. A central corridor is built so the farming area can be accessed easily on foot to replant it, without stepping on the soul sand.
In this design water flows to a central spot the nether wart is harvested with the press of a button, and flows to a central point for easy collection.
Piston powered farms
Piston movement can “shake" the nether wart off to be easily picked up and replanted. In the following video this type of farm is shown, using a simple redstone circuit that triggers the piston to push the soul sand and then to reset the placement.