The North Carolina Museum of Art is quite nice. They house a pretty respectable permanent collection, and host some pretty impressive temporary exhibits, as well (last year's contemporary art and flight exhibit was a personal favorite). They also feature outdoor movie and music performances, as well as some special classes, and there are also a few outside installations, as well as a bike/pedestrian greenway. Admission to the permanent collection is always free, and the special features are always reasonably priced (the movies in the park, for example are only $3). It's certainly worth a walk-through at least once or twice a year.
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