The ongoing growth of three-dimensional (3D) printing has started to expand into the medical field. To date, the main challenge is a lack of information surrounding the materials which accurately mimic soft tissue, and how they can be reproduced into a surgical phantom for medical use. This study reports on successful materials for simulating soft tissue, and the methods in which they can be printed to create surgical phantoms. Noteworthy materials which have been reported in literature as having good concordance with soft tissue mechanical properties have been identified as silicone, gelatin, polyvinyl alcohol (PVA), and Stratasys-manufactured TangoPlus. Four printing techniques, namely material extrusion, material jetting, photopolymerization, and powder bed fusion, are discussed and reviewed on their suitability for fabricating phantoms, and a summary of their use in literature has been presented in tabular format. This study explores the current uses of 3D-printed phantoms for surgical training, surgical planning and patient understanding and discusses ways in which this could be advanced in the future.