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Legilimency
Legilimency, a branch of magic not normally taught at Hogwarts, is the ability to extract emotions and memories from another person's mind. Although the word literally translates as 'mind-reading', its practitioners consider this a naive interpretation of the art. Someone who practices Legilimency is known as a Legilimens.
Legilimency is easier when the spell-caster is physically near the target, and when the target is off-guard, relaxed, or otherwise vulnerable. Eye contact is often essential, so it is useful for a Legilimens to verbally manipulate his or her target into meeting the Legilimens' eyes. This occurs solely on the fringe benefit that the target's emotional state may bring relevant associated memories to the surface.
Voldemort was a highly skilled Legilimens, to the point where Voldemort could nearly always tell when someone was lying to him. Unlike Snape, Voldemort was never shown speaking the incantation to use Legilimency. He was able to possess the bodies of snakes while retaining his own body, possibly through Legilimency. Through his connection with Harry, he was also able to use his ability to feed images and eventually false visions into Harry's dreams, when Harry's mind was most relaxed and vulnerable.
Dumbledore was also a Legilimens, describing himself as sufficiently skilled to know when he was being lied to. He was also sufficiently skilled in Occlumency, to be able to teach it if he so wished.
Since we've been told that the use of Veritaserum is controlled by very strict Ministry guidelines, the same should be true of Legilimency. This would not need to be true of Occlumency, since that is purely a defensive measure and only affects the person using it.)
Teaching Legilimency may be legally restricted. For instance, a student may be required to be of age, and/or to pass character tests like those required for Auror training. Note that nobody suggests Harry should be taught Legilimency at his age, and he isn't learning it as part of Occlumency (as indicated by his thoughts during his History of Magic O.W.L.).
Using Legilimency may or may not be legally restricted. There may be restrictions on the circumstances under which a Legilimens may reveal something learned via the technique, analogous to the confidentiality restrictions that apply to Muggle lawyers, physicians, psychiatrists, and clergy.
Occlumency
Occlumency is the art of magically defending the mind against external penetration, sealing it against magical intrusion and influence - the defensive counter to Legilimency. A practitioner of Occlumency is referred to as an Occlumens.
Occlumency is a necessary prerequisite to defeat a Legilimens' lie-detector abilities without suspicious behavior, such as avoiding face-to-face contact and eye contact. Elementary Occlumency involves clearing the mind of thought and emotion, so that the Legilimens can find no emotional ties to memories that the target wishes to conceal. Simple resistance to attack requires similar skills to those needed to resist the Imperius Curse. In its more advanced form, Occlumency allows the user to suppress only feelings and memories that contradict what the user wishes a Legilimens to believe, thus allowing the Occlumens to lie without self-betrayal.
Occlumency has not often been studied at Hogwarts in the years prior to the second war against Voldemort, particularly not below N.E.W.T. level.
Severus Snape was a particularly skilled Occlumens, since he had managed to survive as a spy among the Death Eaters. This occurred under the nose of Voldemort, a noted skilled Legilimens who had good cause to be suspicious of Snape's loyalty. Dumbledore knew enough Occlumency that he could have taught it if he wished to.
Potions
Potions are magical liquids created by mixing various ingredients in a cauldron, according to very specific rules. These mixtures must usually be drunk to give their magical effect. The ingredients in potions range from the mundane to the bizarre and fantastic, and the procedures for creating some potions can be complicated and time-consuming. Severus Snape was the Potions Master at Hogwarts from 1980 to the fall of 1996. Horace Slughorn took over as Potions Master for the 1996 - 1997 school year. The Potions classroom is located in a chilly, dark dungeon.
Transfiguration
Transfiguration is magic that changes one object into another. It is possible to change inanimate objects into animate ones and vice versa. Some Transfiguration spells alter a part of something, such as changing a person's ears from normal into rabbit ears. At Hogwarts, Professor McGonagall taught Transfiguration. Transfiguration spells were cast in ancient times as well. Circe, a Witch who lived in the Greek island of Aeaea, was famous for turning lost sailors into pigs.
The opposite of Transfiguration is Untransfiguration, which would be returning something to its proper form.
Divinations Divination is magic which attempts to foresee future events. Many in the wizarding world consider this branch of magic to be imprecise at best. There are several types of Divination. The most imprecise is what is commonly known as "fortune telling," and this is what Trelawney taught at Hogwarts. The second kind of Divination is what is practiced by the centaurs. When Firenze started teaching Divination classes in the spring of 1996, he taught these techniques. They were mostly lost on the human students in his classes, though. The third type of Divination is called Seeing. This is true Divination, although what the Seer reveals is usually in the form of a Prophecy, which then itself requires some interpretation. A Seer, one who possesses the Inner Eye, doesn't seem to have control over their Seeing. Trelawney, for example, only made actual Prophecies twice. She made plenty of claims about everything, from troubles ahead for various students to Neville breaking a teacup. However, in each case when she made an actual Prophecy, Trelawney went into a trance and spoke in a completely different voice. Both times, after speaking the Prophecy, she didn't remember a thing about it. True Seeing is very rare, but it has happened plenty of times over the centuries. There is a huge cathedral-sized room of the Department of Mysteries, where records of thousands of Prophecies are stored in glass spheres. Each sphere is labeled with the initials of the Seer who spoke the Prophecy, the person it was spoken to, and with the subject of the Prophecy.