There are no universities, but there is some evidence of "Trade Schools", for lack a better term.
The most mentioned of these is the extensive training that Aurors undergo in order to become Aurors.
The training to become an Auror lasts for 3 years after normal education is complete.
I believe it is reasonable to assume there are other specialty programs like that in existence.
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No there are no Wizarding universities or post-Hogwarts (or Durmstrang, Beauxbatons, Salem Institute, etc) higher eduction.
SOURCE: Harry Potter Lexicon
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Percy goes straight from school (NEWTs - good grades) to working at the Ministry.
This suggests some sort of apprenticeship situation.
His title in book 5 is 'Junior Assistant to the Minister' which he hopes will impress his father - instead his dad says Fudge is probably trying to use him to spy on the family (which is probably true), and this enrages Percy.
Fred and George are mentioned as having mostly good grades 'even though they mess around a lot'
(this is a line in book one, towards the beginning - ron is talking to harry), and they stay in school long enough to do OWLs but not NEWTs.
At 17 they open a joke shop.
Presumably, they did not need an apprenticeship situation, because they are already fairly well-educated and can learn more from working in the shop than from continuing education.
It seems to me that some form of apprenticeship would be the norm, however.
Tonks is mentioned as being mentored by Mad Eye Moody, at the Ministry.
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Rowling doesn't mention any formal Wizarding higher education, which I've always found strange.
But then again these are novels directed at children and centre their coming-of-age from adolescence into early adulthood, so indicating a realistically developed adult society up ahead might detract from their appeal.
(After all, that's Muggle business.)
Hogwarts is clearly based on the UK education system: O.W.L.s are equivalent to GCSEs and N.E.W.T.s are A-Levels, not university degrees.
However, schoolteachers have the title of 'Professor' which gives them the status of real-world academic professionals with advanced degrees...whether they've earned it or not!
("Professor" Lockhart?
Really?)
This makes Hogwarts appear extra special and prestigious to readers, especially as it's one of only three named schools for magic in Europe.
It seems like after school, people can go on to apprenticeship training contracts if they have the right N.E.W.T.s, like what you have to do to become an Auror / Healer / work in particular departments at the Ministry / hang out with dragons in Romania etc.
These are all very practical and hands-on career options.
It makes me wonder what the more cerebral and artistic types do.
It's noticeable that the kids in the novels get married and have children of their own at a really young age after leaving school - considering how long Wizard lifespans are (unless murdered by Voldemort)
it strikes me as odd that they're considered fully grown-up by 18.
(Do they get divorced?
Have casual relationships?
Affairs?
Surely not everyone who survives Voldemort lives monogamously ever after.)
But I suppose it's war, and their society is in crisis and struggling to move beyond its outdated institutions - this is what sets up the plot.
Some of the adult characters are shown to have made huge mistakes when in their late teens and early twenties (e.g. Dumbledore, Snape, the Marauders).
Perhaps a university education could've delayed their terrible decisions and kept them out of trouble for a bit longer :)
It does make me wonder though, just how small or large the Wizarding world really is, and what do the intellectual ones do?
A world without higher education seems limited.
Clearly there are lots of books, but no formal higher academic institutions, and only a handful of magical schools.
Severus Snape in the novels strikes me as a researcher with extensive knowledge.
He doesn't accept less than 'Outstanding' for entry into his N.E.W.T. classes, is clearly an intellectual (very articulate, loner, sneers at 'dunderheads', values subtlety), has a passion for Potions as well as the Dark Arts, and he develops his own spells while still in school.
By his 30s he could probably write a book or two on magic if he chose - and so could Dumbledore (perhaps he did?)
and McGonagall.
I wonder if these Hogwarts experts could have done more for magical education than teach 11-18 year olds, if they weren't also so busy trying to save their society from Voldemort and his 'pureblood' ideology.
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There seems to be apprenticeships and outside study mentioned for many.
In The Philosophers Stone it mentions Dumbledore worked on Alchemy with Nicholas Flamel, presumably before joining Hogwarts.
Hagrid was apprenticed to the previous gamekeeper.
There are mentions of trainee healers in The Order of The Phoenix .
There are certainly gaps in the storylines of the middle aged parents and their education, presumably because the wizarding world was at war in their early 20s and 30s and many families went into hiding.
As in real history it’s not unusual for formal education to take a backseat in times of war or turmoil.
Many people however dont seem to work.
Purebloods seem to just be living off thier inheritances, Sirius, James and Lily Potter, Molly and sadly most of the other mother characters are never mentioned as having a profession.
It ties back to questions about schooling before age 10, do wizard children go to Muggle primary school or are parents busy homeschooling?
They come to Hogwarts at 11 knowing how to read, write, count etc.
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Or is it just 'further education' i.e. Hogwarts, Durmstrung, Beaux-batons? Are there wizarding universities?
Does higher education exist in Harry Potter?
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In  The Philosophers Stone, Dumbledore  is said to have worked on Alchemy with Nicholas Flamel, presumably before joining Hogwarts. There are trainee healers in The Order of The Phoenix and apprenticeships and outside study are mentioned for many. There are no universities though there is evidence of insititutes that can be considered as similar to trade schools. Aside from this, many wizards become apprentices under a more knowledgeable wizzard in their field i.e. Hagrid becoming the apprentice of the previous Games Keeper.
There are a lot of mentions towards apprentice ships and references to go with it i.e. Dumbledore working on alchemy with Nicholas Flamel, Hagrid was apprenticed by the previous gamekeeper etc. There are no wizarding universities though there is evidence of "Trade Schools" for lack of a better term.