What do universities usually do with a student's graded final exams?

Speaking as a professor, I typically hold on to them for a year or so just in case any student wishes to see them. That said, I can remember this happening only once in about 5000 students taught.

When I was in college, we had the option of providing self-addressed stamped envelopes in which our final exams could be returned to us. Over half of the students availed themselves of this option.

I do not believe that my university has any formal policy, though they could have one that I am unaware of. It would seem foolish for a professor to get rid of the exams before the students have had an opportunity to see their grades and respond if something seems out of line.

At our university, the lecturer has to store your exams (along with the original test and answer key) for at least 2 years. This is so that a committee has the possibility of looking at them during an audit, to determine whether the exam was of sufficient level and graded properly. Usually such an audit does not take place and the exams are simply discarded once the lecturer runs out of shelf space.

There is a 30 day period in which students can request a copy of their exams. After these 30 days, even though the exams are stored, they are no longer entitled to this - the storage is purely for the eventuality of an audit.

An out-of-the-way corner of my office contains stacks of final exams from the last few courses I've taught. Students are welcome to stop by my office and retrieve their exams; very few do so, but more than the one in 5000 mentioned by Corvus. We're not supposed to discard the exams until some time (maybe a year, maybe 3, I don't remember offhand) has passed, presumably because students might open a grade appeal process. I generally keep exams longer, but not forever; when that out-of-the-way corner gets too full (and when I notice it) I weed out the oldest exams.