Tissue engineering (TE) is a branch of regenerative medicine with enormous potential to regenerate damaged tissues using synthetic grafts such as scaffolds. Polymers and bioactive glasses (BGs) are popular materials for scaffold production because of their tunable properties and ability to interact with the body for effective tissue regeneration. Due to their composition and amorphous structure, BGs possess a significant affinity with the recipient’s tissue. Additive manufacturing (AM), a method that allows the creation of complex shapes and internal structures, is a promising approach for scaffold production. However, despite the promising results obtained so far, several challenges remain in the field of TE. One critical area for improvement is tailoring the mechanical properties of scaffolds to meet specific tissue requirements. In addition, achieving improved cell viability and controlled degradation of scaffolds is necessary to ensure successful tissue regeneration. This review provides a critical summary of the potential and limitations of polymer/BG scaffold production via AM covering extrusion-, lithography-, and laser-based 3D-printing techniques. The review highlights the importance of addressing the current challenges in TE to develop effective and reliable strategies for tissue regeneration.