Over the past few decades, additive manufacturing (AM) has become a reliable tool for prototyping and low-volume production. In recent years, the market share of such products has increased rapidly as these manufacturing concepts allow for greater part complexity compared to conventional manufacturing technologies. Furthermore, as recyclability and biocompatibility have become more important in material selection, biopolymers have also become widely used in AM. This article provides an overview of AM with advanced biopolymers in fields from medicine to food packaging. Various AM technologies are presented, focusing on the biopolymers used, selected part fabrication strategies, and influential parameters of the technologies presented. It should be emphasized that inkjet bioprinting, stereolithography, selective laser sintering, fused deposition modeling, extrusion-based bioprinting, and scaffold-free printing are the most commonly used AM technologies for the production of parts from advanced biopolymers. Achievable part complexity will be discussed with emphasis on manufacturable features, layer thickness, production accuracy, materials applied, and part strength in correlation with key AM technologies and their parameters crucial for producing representative examples, anatomical models, specialized medical instruments, medical implants, time-dependent prosthetic features, etc. Future trends of advanced biopolymers focused on establishing target-time-dependent part properties through 4D additive manufacturing are also discussed.