The Duchy of Nuts is an area of the Land of Ooo. It is the home of the Duke of Nuts, his family, and other Nut People. Everything here, including birds and trees, are nut-themed. It is a duchy (or dukedom), not a kingdom because it is ruled by a duke and duchess. It is unknown whether the duchy is an independent territory or a subordinate province within the Peanut Kingdom ruled by Peanut Princess.
The Duke of Nuts' castle is a small structure owned by the Duke and Duchess of Nuts. It is smaller than an average castle and more like a manor. There is a grand hall of some sort where they celebrate the Duke's second son's first birthday in "Henchman." Lisby the Cashew Butler appears to guard the front door, along with several other Nut People.