An often overlooked sight in Berlin is the Spandau Citadel. In German “Castle” is more of a “Palace”, but this seems to be a common misconception for most of us. This is a real castle, not for kings but for defense of the city. It is pretty large and has a giant tower to its side. The Castle and Tower can be accessed at the main gate, but inside the walls are free to roam.
Inside the walls you can find several art galleries and Museums, as well as an art school. Inside the Castle you can roam a small museum and walk across the city rampants, but I think it is possible to do all of this without the Museum by going across the Italian Courtyard and walking up to the tower.
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