1. In the place where you belong; in your own personal surroundings; apart from other people. They are in a little world of their own in their house on the mountain. 2a. In deep thought or concentration. Mary is in a world of her own when she is playing the piano.
Compare: LOSE ONESELF. 2b. slang Not caring about or connected with other people in thoughts or actions. Usually used sarcastically. That boy is in a world all by himself. He never knows what is happening around him.