Additive manufacturing of biopolymer composites is rapidly advancing biomedical engineering. By combining synthetic polymers with natural biopolymers, inorganic ceramics, nanomaterials, and bioactive molecules, AM enables patient-specific constructs with high geometric precision and tunable biological performance. These materials support applications across hard and soft tissues, including bone regeneration, cartilage repair, wound healing, vascular grafts, and cancer modeling. Despite this progress, major barriers remain: mechanical-degradation mismatches, limited vascularization in thick constructs, photoinitiator-related cytotoxicity, and inter-laboratory variability that reduces reproducibility and slows clinical translation in real-world settings. Broader adoption is further constrained by the lack of standardized protocols and clear regulatory pathways. We compare AM modalities (extrusion, vat photopolymerization, and powder-bed processes) through a cost-benefit lens, outlining when each optimally balances resolution, throughput, sterility, and cell compatibility. To address these gaps, recent work (2020–2025) is shifting the field from static scaffolds toward adaptive, intelligent, and sustainable platforms, leveraging 4D printing, nanotechnology-enabled reinforcement and bioactivity, AI-driven design optimization, and greener feedstocks such as sustainable biopolymers. Notably, 2025 reports on multi-material bioprinting for neural tissues show improved print fidelity and bioactivity, enabling more realistic microvascular networks and neural interfaces. By synthesizing these advances and critically evaluating current limitations, this review proposes strategic directions for AM to mature into a clinically relevant, globally sustainable manufacturing approach. Biopolymer composites are not incremental upgrades; they are pivotal enablers of next-generation regenerative medicine.