Additive Manufacturing (AM), also known as 3D printing, has been highlighted as a complementary method to the traditional (subtractive and formative) manufacturing. This mainly results from its distinctive characteristics to directly produce complex shapes and assemblies without an assembly process. With these aspects, AM has affected the way products are designed and formed, which leads to an exclusive research area, known as Design for AM (DfAM). As a step towards addressing DfAM, this paper reviews the literature on re-designing an original model into assemblies produced in AM, named as Part Decomposition (PD). Although PD has received less attention in DfAM compared with Part Consolidation (PC) that is re-designing assemblies into a consolidated single part, PD has been studied with various motives and challenges for AM. To investigate the research trend in PD, 37 main publications are categorized under five motives including printability, productivity, functionality, artistry and flexibility. Additionally, from technical and methodological aspects, relevant studies are organized into decomposition issues (automatic, semi-automatic and manual decompositions), buildup issues (orientation decision for single- and multi-part and packing problem), and assembly issues (connection design and assembly process planning). As witnessed in this comprehensive review, the concept of PD leaves further research challenges spanning several disciplines. Along this line, we further elaborate future research directions of PD under three main categories: (1) enhancing the AM productivity for mass customization; (2) developing novel decomposition methods and guidelines; and (3) applying conventional design methodologies to PD.