Three-dimensional (3D) printing is a game changer technology that holds great promise for a wide variety of biomedical applications, including ophthalmology. Through this emerging technique, specific eye tissues can be custom-fabricated in a flexible and automated way, incorporating different cell types and biomaterials in precise anatomical 3D geometries. However, and despite the great progress and possibilities generated in recent years, there are still challenges to overcome that jeopardize its clinical application in regular practice. The main goal of this review is to provide an in-depth understanding of the current status and implementation of 3D bioprinting technology in the ophthalmology field in order to manufacture relevant tissues such as cornea, retina and conjunctiva. Special attention is paid to the description of the most commonly employed bioprinting methods, and the most relevant eye tissue engineering studies performed by 3D bioprinting technology at preclinical level. In addition, other relevant issues related to use of 3D bioprinting for ocular drug delivery, as well as both ethical and regulatory aspects, are analyzed. Through this review, we aim to raise awareness among the research community and report recent advances and future directions in order to apply this advanced therapy in the eye tissue regeneration field.