I might also add:
The University is a semi-autonomous agency of the government, i.e. it gets funding and certain guidelines from the state, but can set its own admittance requirements, etc.. UT Austin has the largest student body of any university in the United States, which places it "high in the running for" largest "world-wide." This, though, is due to several factors. Some of the lowest tuition rates make the school highly desirable (see above). Also, quite a large portion of Texans are "raised" Burnt Orange or Maroon. That is to say, their parent or parents went to either UT or A&M and so have filled their homes and thus their child's consciousness with school pride for their alma mater. I know several people who have "always wanted" to go to UT. Many people consider the size of the University to be a bad thing, but I've found it's not a big deal. In the classes where it counts the numbers are small, in my experience and in the classes that are huge either discussion wouldn't really fit or you just have to speak up (I have an Intro to Theatre class with over one hundred people and class discussion was thought provoking and interesting, even at 8:00 AM).
A few other (random) facts about UT Austin:
Information courtesy of various and sundry parts of www.utexas.edu and my own experience.