The case hinges on whether students applying to public universities should be given preference based on their race.
UT currently uses an admission system that looks at a variety of factors, including race, to determine whether to accept a student.
Law Professor Charles "Rocky" Rhodes, with the South Texas College of Law, says this is the most significant case on the court's agenda right now.
"If the Supreme Court essentially said that race cannot be considered at all as a factor anymore in the admissions program, of course that's going to make a huge impact on elite public universities."
A ruling against UT in this case, could have a chilling effect on affirmative action. Schools often use race as a determining factor in order to achieve diversity on their campuses.
Texas schools are already required to accept the top ten percent of graduating classes as a means to that end.
"And so that's going to be one of the arguments, that there are other ways to do this. You don't have to continue to consider race. And the last time the Supreme Court considered, when they did so in 2003, Justice O'Connor's opinion said that race-based preferences have to be limited in time, that they can't go on and on forever."
That was less than ten years ago, so the question is will the Supreme Court think that time limit has expired. Some states, including California and Michigan, already have laws prohibiting public universities from considering race in the admissions process.