The advent of additive manufacturing offered the potential to revolutionize clinical medicine, particularly with patient-specific implants across a range of tissue types. However, to date, there are very few examples of polymers being used for additive processes in clinical settings. The state of the art with regards to 3D printable polymeric materials being exploited to produce novel clinically relevant implants is discussed here. We focus on the recent advances in the development of implantable, polymeric medical devices and tissue scaffolds without diverging extensively into bioprinting. By introducing the major 3D printing techniques along with current advancements in biomaterials, we hope to provide insight into how these fields may continue to advance while simultaneously reviewing the ongoing work in the field.