The National Art Center Tokyo (NACT) boasts the country’s largest exhibition space for visiting shows, usually from prestigious international institutions like Paris' Musée de Cluny or Vienna's Kunsthistorisches Museum. It also curates its own shows, often on contemporary Japanese art.
No matter what's on, the building itself is spectacular, with an awesome undulating glass facade designed by influential architect Kurokawa Kishō. It's free to enter; you only have to pay for admission to the galleries. Take advantage of the bright, airy space and have a coffee at one of the museum's two cafes, casual Café Coquille in the atrium lobby or the fancier Salon de Thé Rond, perched on an inverted glass cone.
NACT is open daily from 10am to 6pm, except Tuesdays. While you're here, head to the basement to check out the excellent gift shop, Souvenir from Tokyo, where you can find an ever-changing selection of covetable souvenirs from local designers.