Additive manufacturing in medicine is moving from passive structural replacements toward active, biologically interactive materials that can respond, regenerate, or gradually dissolve after fulfilling their function. This review summarizes studies from 2020–2025 covering four classes of emerging materials: (i) smart and stimuli-responsive systems, including shape-memory polymers and hydrogels that change form or properties in response to heat, acidity, or moisture; (ii) regenerative bioinks and hydrogels, both natural and synthetic, developed to enhance printability, cell survival, and vascular growth; (iii) bioresorbable metals and composites, such as magnesium and zinc alloys, processed by additive manufacturing and modified to maintain strength while controlling the rate of degradation inside the body; and (iv) nano-functionalized feedstocks in which nanoparticles of silver, copper, zinc oxide, graphene, or hydroxyapatite provide antibacterial, electrical, or bone-stimulating functions. The paper discusses the balance between printability and biocompatibility, the stability of new materials during sterilization, and their mechanical durability compared with biological tissues. Regulatory aspects for variable additive manufacturing batches and biologically active materials are also reviewed. The discussion concludes with three guiding questions for clinical adoption: Is it printable? Is it safe? Is it superior to conventional titanium or polyetheretherketone implants? Bioresorbable magnesium systems and smart polymer platforms appear closest to near-term clinical use as process control and material standards advance.