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Greek life, varsity, club, and intramural sports, clubs, volunteering, and the student government as well as its various committees are very popular on campus. I don't think I've met someone who isn't involved in one of the things above. As you can expect, athletic events are HUGE on campus, particularly football and basketball games. The Student Government always has comedians and guest speakers (this week we had Billy Joel!) so if that's more your speed there are events for you!
On the weekends, you'll see students heading over to Carriage Apartments, Huskies Bar, Ted's Bar, or Thirsty's Bar. if you aren't into going out, the Student Union has a late night every friday that has free giveaways and karaoke; it's actually really popular!
Basically, a UConn student works hard during the week then parties hard Thursday-Saturday.