Polymer 3D printing is an emerging technology with recent research translating towards increased use in industry, particularly in medical fields. Polymer printing is advantageous because it enables printing low-cost functional parts with diverse properties and capabilities. Here, we provide a review of recent research advances for polymer 3D printing by investigating research related to materials, processes, and design strategies for medical applications. Research in materials has led to the development of polymers with advantageous characteristics for mechanics and biocompatibility, with tuning of mechanical properties achieved by altering printing process parameters. Suitable polymer printing processes include extrusion, resin, and powder 3D printing, which enable directed material deposition for the design of advantageous and customized architectures. Design strategies, such as hierarchical distribution of materials, enable balancing of conflicting properties, such as mechanical and biological needs for tissue scaffolds. Further medical applications reviewed include safety equipment, dental implants, and drug delivery systems, with findings suggesting a need for improved design methods to navigate the complex decision space enabled by 3D printing. Further research across these areas will lead to continued improvement of 3D-printed design performance that is essential for advancing frontiers across engineering and medicine.