This paper presents a thorough examination of recent progressions in additive manufacturing (AM) methodologies utilized within the biomedical realm, with particular attention to tissue engineering. Additive manufacturing provides exceptional precision in producing intricate three-dimensional objects using a variety of materials. Techniques such as Jetting, Material Extrusion, Material Jetting, Powder Bed Fusion, Sheet Lamination, and Vat Polymerization are strategically employed in biomedical applications to fulfill specific component requirements. Our focus is on polymer materials, encompassing both natural and synthetic variants, exploring the use of hydrogels for scaffold fabrication. We critically analyze the mechanical properties of these polymer scaffolds to enhance personalized patient care and mitigate implantation risks. Through careful adjustment of process parameters, this study illustrates the feasibility of achieving improved mechanical properties in manufactured components. This comprehensive review contributes to the ongoing discussion on advanced manufacturing methods for polymeric scaffolds in medical contexts, offering valuable insights for researchers, practitioners, and industry experts.