* The full name: University of California, Berkeley; Harvard University; Massachusetts Institute of Technology (MIT)
* Abbreviations: UCLA; MIT; USC
You do *not* capitalize the generic word "university" unless it's part of a proper noun:
* Correct: I attended the University of Michigan.
* Correct: She's applying to several universities.
* Incorrect: I attended the University Of Michigan. (unless it's a stylized logo)
* Incorrect: She's applying to Several Universities.
In short, capitalize the official name, but not the general term "university".