This paper presents Motion Puzzle, a novel motion style transfer network that advances the state-of-the-art in several important respects. The Motion Puzzle is the first that can control the motion style of individual body parts, allowing for local style editing and significantly increasing the range of stylized motions. Designed to keep the human's kinematic structure, our framework extracts style features from multiple style motions for different body parts and transfers them locally to the target body parts. Another major advantage is that it can transfer both global and local traits of motion style by integrating the adaptive instance normalization and attention modules while keeping the skeleton topology. Thus, it can capture styles exhibited by dynamic movements, such as flapping and staggering, significantly better than previous work. In addition, our framework allows for arbitrary motion style transfer without datasets with style labeling or motion pairing, making many publicly available motion datasets available for training. Our framework can be easily integrated with motion generation frameworks to create many applications, such as real-time motion transfer. We demonstrate the advantages of our framework with a number of examples and comparisons with previous work.