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The Hogwarts School of Witchcraft and Wizardry is a fictional magic school that is the main setting of the Harry Potter novels by J. K. Rowling. There are several teachers (called professors), each specialising in a single subject. Some are required, others electives. Pupils get a chance to sign up for the latter kind towards end of their second year, and start them at the beginning of their third year. It seems that some can be dropped at the end of the fifth.
The art of metamorphosis, turning something into something else by means of waving one's wand and saying magic words. One of the trickier subjects, it requires great concentration and effort. Transfiguration is taught from the first year to the seventh.
Before Albus Dumbledore became headmaster, he served as Transfiguration teacher. The current Transfiguration teacher is Minerva McGonagall, who's a competent teacher and a stern but fair disciplinarian.
The use of spells, potions and other techniques to counteract the Dark Arts, demons and evil creatures such as the Boggart. Defence Against the Dark Arts is taught from the first year to the seventh.
The position of teacher of Defence Against the Dark Arts is coveted by Hogwarts' Potions master Severus Snape, although he has yet to receive it, possibly due to his prior association with Lord Voldemort. When asked about the reason Dumbledore does not let Snape have the job, the author responded that Dumbledore believes that teaching Defence Against the Dark Arts would bring out Snape's worse side(!).
In each of the first five books of the series, the current Defence Against the Dark Arts teacher plays a central role in the plot; each is either replaced, forced to resign, killed, or driven insane by the end of the relevant book. The quality of teaching varies widely from teacher to teacher.
Spoiler warning: Plot or ending details follow.
Defence Against the Dark Arts Instructors have included:
The art of moving and manipulating objects by means of waving one's wand and saying magic words. This class is one of the noisier ones; a group of students practicing how to make objects move towards them or how to cause people to laugh makes a lot of hubbub even when they're doing well, never mind when they misfire a lot. Charms is taught from the first year to the seventh.
Filius Flitwick is the current Charms teacher. J. K. Rowling says if she were to teach at Hogwarts, she would teach Charms.
The art of creating potions with magical effects. The process is delicate and instructions must be followed exactly in order to achieve the desired result. Potions is taught from the first year to the seventh.
Severus Snape is the current Potions master, although he covets the Defence Against the Dark Arts job. Although a good teacher, he's not the fairest nor the most pleasant of the lot. Students of his house are often quite like his classes, while those of other ones end up intimidated and/or angry.
Very similar to its real-world equivalent. Astronomy classes take place in the tallest tower and occur every Wednesday. Astronomy is taught from the first year to the seventh. By their fifth year, students are expected to be able make a star chart.
Professor Sinistra is the current Astronomy teacher.
The use of enchanted broomsticks. Flying lessons are presumably only given to first-year students.
Madam Hooch is currently teaching flight.
The study of magical history. History of Magic is taught from the first year to the seventh.
The current History of Magic teacher, Professor Binns, isn't very effective. He is the only ghost teacher, extremely set in his ways; some say he didn't even notice he was dead, he just got up one day in the staff room, leaving his body behind.
Binns spends his class drawling on and on, and does not seem to mind (or notice) that the class is never paying any attention.
The study of magical plants (such as mandrakes), and how to take care of, utilize and/or combat them. Herbology is set in greenhouses on the grounds. First Years only work in Greenhouse One, as Greenhouse Three contains more dangerous plants. Classes start using Greenhouse Three in their second year.
Professor Sprout is the current Herbology teacher.
A branch of magic that's concerned with the magical properties of numbers; not taken by the protagonist, nor by Ron Weasley (though the favourite subject of Hermione Granger), lessons on the subject are not shown and their precise contents remain unknown. The only information given is that the subject is quite hard - therefore attractive to Hermione - and running parallel to Divination lessons at Hogwarts. An optional subject, taught from the third year to the seventh.
Professor Vector is the current Arithmancy teacher and has yet to have a speaking part.
Self-explanatory. No details are known, for the same reasons as Arithmancy.
Predicting the future. Various methods are involved, including crystal balls, palmistry, tea leaves and the interpretation of dreams.
The supporters of the subject claim that it's a fairly inexact science that needs innate gifts more than bookwormish tendencies. Those opposed to the subject claim that it's a load of poppycock and that the teacher is nothing but an old fraud. The latter kind includes Minerva McGonagall and Hermione Granger Professor Dumbledore even considered dropping Divination from the curriculum altogether, but that changed when he interviewed Sybill Trelawney. An optional subject, taught from the third year to the seventh.
The current Divination teacher is Sybill Trelawney. When Dolores Umbridge fired her, she was replaced (to Umbridge's fury) with Firenze the centaur. Presumably Trelawney got her job back after Umbridge left.
Leearning about and to care for magical beasts. Classes are held outside the castle. An optional subject, taught from the third year to the seventh.
At the end of the 1993 school year, the Care of Magical Creatures teacher, Professor Kettleburn, quit because he wanted to enjoy some time with his remaining limbs. As groundskeeper Hagrid had been cleared of opening the Chamber of Secrets, Dumbledore gave him the Care of Magical Creatures job. When he is gone he is replaced by Wilhelmina Grubbly-Plank. Hagrid is rather inexperienced but learning fast; his main problem is an inability to grasp the concept of "terribly dangerous monster".
The study of muggles, non-magical people, which involves writing essays such as Why Muggles Need Electricity. Muggle Studies is considered a soft option, but Percy Weasley thinks it is important for a wizard to have an understanding of the non-magical community, especially if he works closely with them. An optional subject, taught from the third year to the seventh.
The art of shielding the mind against external penetration. Definitely not a part of the usual curriculum.
Spoiler warning: Plot or ending details follow.
In Harry's fifth year, Dumbledore arranged for Harry to take one-on-one occlumency lessons with Professor Snape, though Snape told his students that Harry was taking remedial Potions lessons.