There is one ongoing exhibition called Common Ground which is about Japanese American history from early immigration through the WWII incarceration to reparations and redress in the 80's. Other than that which is up all the time, rotating exhibitions change about 4-6 times a year. Check janm.org for current information about the exhibition lineup.
On May 19th they're opening another exhibit What We Carried: Fragments & Memories from Iraq & Syria that will be open till August 5th.
The internment camp exhibit is so large and beautiful upstairs, I wonder if that one is permanent? I know the downstairs exhibition on the camps closed this past Sunday.
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