If you are talking casually about a school that has a longer formal name, then no, you would not capitalize. It's possible that there is a club or restaurant somewhere named "The University", in which case you would capitalize.
If that is the course title, you do capitalize it.
Yes, you should capitalize "Chess Club" as it is a specific name or title of a club.
You should capitalize Colosseum when referring to the specific ancient Roman amphitheater in Rome.
Yes, you should capitalize "The Rotunda" as it is a specific place or location.
You should capitalize "Musical Theatre" when referencing the specific genre or art form.
When you are talking about a specific university, you should capitalize it as a proper noun. When you are talking about universities as place, you can leave it lowercase. Which university does she go to? She goes to Stanford University.
If that is the course title, you do capitalize it.
Campus of the University of Montana was created in 1893.
No, the Princeton University campus is in Princeton, NJ USA.
No there is not a University of Minnesota campus in California.
Hansung University Design campus was created in 2001.
Ohio University Eastern Campus was created in 1957.
The motto of Campus of the University of Montana is 'Lux et Veritas'.
Macquarie University Campus Experience was created in 2008.
GITAM University Hyderabad Campus was created in 2009.
University Campus Suffolk was created on 2007-08-01.
University of Nottingham Malaysia Campus was created in 2000.