Three-dimensional printing (3DP) technologies are characterized as a set of innovative manufacturing techniques that allow for the creation of complex and/or personalized three-dimensional physical objects on the work surface of a 3D printing machine (based on the computer-aided design (CAD) project designs of these parts). Three-dimensional printing techniques are widely used in various areas of knowledge, such as education, engineering, and biomedicine. Polymeric materials are widely used for these applications, mainly due to their desirable workability during part manufacturing, compatibility with other chemical materials, the wide range of polymers with different physical and chemical characteristics, and the possibility for recycling. The development of polymeric drug delivery systems (DDSs) by 3D printing is currently an active field of research, both in academia and industry, given the potential of this technique for medical purposes. In this context, this work reviews potential polymers for the production of drug delivery systems via 3D printing techniques. The demonstrations of the main 3DP techniques used for drug delivery applications include their working principles and advantages and how the technologies develop the final product. In addition, potential synthetic and natural polymers that are currently used in 3DP drug delivery devices are presented and discussed based on recent scientific studies.