Continuity mistake: When Anna meets the King's children for the first time, she is handed a pink rose which she tucks into her belt. When she visibly changes her mind about leaving and walks towards all the children with her hands out, the rose is gone. Next shot, it's tucked back into her belt again.
Factual error: When speaking to a member of the the Royal Family in Siam (modern day Thailand), one have to speak "rachasap" or Royal Thai. The movie depicts characters speaking colloquial Thai to the King.
Continuity mistake: During the 'Shall We Dance' scene, Anna takes off her shawl and places it on the foot stool she is sitting on. When the King jumps onto the stool mid dance it has disappeared.