Tissue engineering has become a central discipline within regenerative medicine, yet critical gaps persist between laboratory innovation and clinical translation. This review systematically appraises advances published from 2003 to 2025, focusing on three mechanistically interconnected pillars: (1) scaffold design and biomaterial innovation—including collagen-based matrices, synthetic biodegradable polymers, decellularized extracellular matrix (ECM) scaffolds, and stimuli-responsive hydrogels; (2) biofabrication technologies, with mechanistic analysis of three-dimensional (3D) and four-dimensional (4D) bioprinting modalities, artificial intelligence (AI)-driven scaffold optimization, and organ-on-a-chip systems; and (3) translational applications across cardiovascular, musculoskeletal, craniofacial, and urological systems. Literature was selected through a structured search of PubMed, Web of Science, and Scopus using terms including "tissue engineering", "scaffold", "bioprinting", "decellularization", "biomaterials", and "clinical translation", supplemented by manual citation tracking of landmark studies. We identify three persistent research gaps that the field has not adequately resolved: the vascularization of constructs exceeding the oxygen-diffusion limit (~200 µm), the absence of validated surrogate endpoints for regulatory clinical trials, and the reproducibility deficit in 4D bioprinting and AI-designed scaffold studies. Critical comparison of competing approaches—collagen versus synthetic polymer scaffolds, extrusion versus stereolithographic bioprinting, autologous versus allogeneic cell sources—reveals trade-offs not captured in descriptive reviews. By contextualizing mechanistic detail within translational reality, this review offers a resource for researchers seeking not only what has been achieved, but why specific bottlenecks remain and how converging technologies may resolve them.