Material Extrusion Additive Manufacturing (MEAM) is revolutionizing rapid prototyping to create complex structures and detailed surfaces. Traditional three-axis 3D printers are limited by planar slicing, restricting the ability to achieve solid freeform fabrication (SFF) due to confined print directions. In contrast, non-planar slicing, also known as curved layer slicing, allows printing in an unlimited range of directions. This innovation necessitates multi-axis 3D printers with more than three degrees of freedom (DOFs), enabling a more agile fabrication process. Despite these advancements, there is a notable gap in systematic research on the machine, mechanisms, and process planning involved in non-planar printing, particularly for printers that operate with collaborative modules (refers to a left-and-right hand system). Thus, this paper critically analyzes the topological configurations of current 3D printing equipment and investigates the application of Quotient Kinematics Machines (QKMs) for innovative machine design. Process planning is vital to achieving fine control and additive processes. Therefore, slicing and path planning strategies (both explicit and implicit) for non-planar printing are reviewed. Composite material MEAM is also discussed, including maximizing anisotropy in continuous-fiber-reinforced polymer (CFRP) parts and 4D planar printing for morphing structures. Moreover, this review addresses the challenges and opportunities surrounding the development of reconfigurable 3D printers, the importance of intelligent planning and in-situ MEAM, and the overarching need for integrated manufacturing, monitoring, and control. Ultimately, the paper highlights the transformative impact that advanced machine design, mechanism synthesis, and process planning strategies could have on MEAM, particularly in achieving versatile print orientations and advanced material deposition to push the boundaries of what can be manufactured.