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A nice aquarium that features a dolphin show and a transparent underwater tunnel.
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棟方志功記念館
Shiko Munakata is famous for his woodblock prints, displayed in this very small gallery along with some of his paintings and calligraphy. Closed Mondays except for national holidays and during Nebuta.
青森県立美術館
Exhibits about the natural and cultural history of Aomori prefecture.
アスパム
A large triangular building that can be easily spotted from the train station or the bridge, this is the tourism center for all of Aomori prefecture. It includes a movie about the prefecture (Japanese only) and an observation deck with views of the harbor and city (extra charge).
善知鳥神社
There are nice gardens and ponds tucked away behind the main shrine. There's also a "power spot" -- it's said if you wash your coins in a certain fountain there, you'll be free from money troubles.
八甲田丸
The ferry which connected Hokkaido and Aomori prior to the building of the Seikan Submarine Tunnel is now a somewhat run down museum, but worth checking out if you're interested in regional history, trains, or boats. The rear part of the deck of the ferry serves as a beer garden on summer nights.
青森ベイブリッジ Aomori Bei Buridji
This bridge spans a short harbor inlet near Aomori Station and the former ferry port.