Additive manufacturing (AM, also known as 3D printing) is an advanced manufacturing technique that has enabled progress in the design and fabrication of customised or patient-specific (meta-)biomaterials and biomedical devices (e.g., implants, prosthetics, and orthotics) with complex internal microstructures and tuneable properties. In the past few decades, several design guidelines have been proposed for creating porous lattice structures, particularly for biomedical applications. Meanwhile, the capabilities of AM to fabricate a wide range of biomaterials, including metals and their alloys, polymers, and ceramics, have been exploited, offering unprecedented benefits to medical professionals and patients alike. In this review article, we provide an overview of the design principles that have been developed and used for the AM of biomaterials as well as those dealing with three major categories of biomaterials, i.e., metals (and their alloys), polymers, and ceramics. The design strategies can be categorised as: library-based design, topology optimisation, bio-inspired design, and meta-biomaterials. Recent developments related to the biomedical applications and fabrication methods of AM aimed at enhancing the quality of final 3D-printed biomaterials and improving their physical, mechanical, and biological characteristics are also highlighted. Finally, examples of 3D-printed biomaterials with tuned properties and functionalities are presented.