Additive manufacturing (AM) has revolutionized the field of biopolymers by providing a range of opportunities to create complex and intricate structures. Here we provide an overview of developments in biopolymer AM, first in the context of how natural systems have impacted the synthetic processes of AM, and secondly in the context of how synthetically derived structures can be used for biomedical applications. In particular, we explore the range of biopolymers developed for AM, AM techniques for biopolymers, and potential biocompatible medical applications with a focus on four categories of tissue in the human body: skin, bone, heart, and nerve. Despite the current challenges of the technology, exciting results in tissue regeneration, synthetic replacements for natural grafts, and complex dynamic organoids have already come to pass - with many more possible in the future.