The Castle is okay but its not the original one as that was burnt down.
The Contemporary Gallery is fab and I really liked it when I was there a couple of weeks ago.
Old favourites are the Caves, Wollaton Hall and Park, have a pint in Ye Olde Tripe to Jerusalem, go and watch an ice hockey game, Goose Fair if it is on, cricket at Trent Bridge, a football game at either Notts Forest or Notts County grounds, take a walk along the River Trent etc.
Nottingham is an okay city with a very busy nightlife - especially when the students are in town. It also has the reputation for one of the roughest cities in the UK - lots of drug related crimes and lots of drinking issues. over 6 years ago