Craniofacial tissues are organized with complex 3-dimensional (3D) architectures. Mimicking such 3D complexity and the multicellular interactions naturally occurring in craniofacial structures represents one of the greatest challenges in regenerative dentistry. Three-dimensional bioprinting of tissues and biological structures has been proposed as a promising alternative to address some of these key challenges. It enables precise manufacture of various biomaterials with complex 3D architectures, while being compatible with multiple cell sources and being customizable to patient-specific needs. This review describes different 3D bioprinting methods and summarizes how different classes of biomaterials (polymer hydrogels, ceramics, composites, and cell aggregates) may be used for 3D biomanufacturing of scaffolds, as well as craniofacial tissue analogs. While the fabrication of scaffolds upon which cells attach, migrate, and proliferate is already in use, printing of all the components that form a tissue (living cells and matrix materials together) to produce tissue constructs is still in its early stages. In summary, this review seeks to highlight some of the key advantages of 3D bioprinting technology for the regeneration of craniofacial structures. Additionally, it stimulates progress on the development of strategies that will promote the translation of craniofacial tissue engineering from the laboratory bench to the chair side.