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Qual questions seem like a great way to study for a new topic, since they usually test slightly deeper understanding than typical questions in a textbook. Princeton has this great archive of questions here:

http://www.math.princeton.edu/generals/topic.html

I was wondering if anyone knows of other universities with similar archives? I'm especially interested in questions from the special topics exam which is usually given orally to Ph.D. students in their 2nd to 3rd year. These tend to ask broader questions than the written exam taken during the first year and trying to answer these questions usually improves one's general understanding of a field.

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    @pki: you may be interested in this other question where I have just compiled a long list of links to qualifying examinations of U.S. departments: http://math.stackexchange.com/a/270467/40582013-01-04

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Harvard has one, Cambridge Tripos used to be harder (so I heard), Purdue also has some posted, and University of Florida's Math Department essentially answered your question (by copying from Dave Renfro) before you even asked by listing qual exams of other universities (UF's page seems to have disappeared completely; anyone who knows the replacement is free to edit).

And then there is Google...

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    Links are dead...2016-06-10
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My alma mater has an an archive of them here. They don't have two different tiers of exams, but rather two pass levels: Master's and PhD (or, at least that's how it worked when I was there).