History of Mudge
Mudge is a sandy, desert-like area in the south-east corner of Oz. It's inhabitants have blue beards. In times before Ozma came to the throne, the Mudgers would often pillage and ransack neighboring communities in Munchkin Country. Sometime after Ozma came to power, Glinda the Good flew over the kingdom in her swan chariot and dropped a book into the kingdom, which carried the threat that if any Mudger were to leave Mudge, they would be beheaded by order of Princess Ozma. After this, the Mudgers would resort to stealing from each other, rather than from other Munchkins.
The one way to get to Mudge, which was accidentally discovered by the American clown Notta Bit More, is to say the rhyme "Udge, Budge, go to Mudge! Udger, Budger, you're a Mudger!" The book doesn't quite explain how this spell was created, but it was likely something created in response to the Mudger invasions to send them home.
Mudge is ruled over by Mustafa, a sultan-like ruler who is obsessed with capturing lions.
Ruth Plumly Thompson based Mudge on an unflattering depiction of Arabs, and Mudge may also have been based on the Ottoman Empire (Mustafa being a Turkish name). This is unfortunately par the course for many of her books. Ozma's threat to behead any Mudger for leaving Mudge is another example of how Thompson's Ozma is a bit more ruthless and merciless than Baum's original characterization (another such example is when she has Mombi executed in a later book).