Natural biopolymers have been widely employed as biomaterial ink hydrogels for three-dimensional (3D) extrusion bioprinting in the preparation of the next generation of bioengineering materials for healthcare applications. Alginate is a linear anionic polysaccharide with favourable properties, such as: typical rheological (gelling, viscosifying, and stabilizing dispersions) characteristics, biodegradability and biocompatibility properties. However, in order to improve alginate applicability for practical biomaterial/bio ink for advanced medical applications, it is often modified and functionalized with several polymers and nanomaterials in order to obtain better printability of alginate-based biomaterial/bio ink hydrogels. This review, principally, emphasizes the recent developments and with a comprehensive overview of alginate-based biomaterial/bio ink hydrogels and their biomaterials (3D scaffolds, tissue-like structures with hierarchical vasculatures, mimics of biological, physiological and pathological functionalities) for biomedical applications. It also addresses the significance of alginates, oxidized alginate and their functionalizations (interface) with various materials in order to improve the biomaterial/bio ink properties for 3D extrusion bioprinting applications. Finally, it provides current advances, vital roles and new perspectives of alginate-based materials and their future developments for 3D bioprinting purposes.