The dawn of 3D printing in medicine has given the field the hope of vitality in many patients fighting a multitude of diseases. Also entitled bioprinting, this appertains to its sequential printing of precursor ink, embodying cells and polymer/composite in a predetermined trajectory. The precursor ink, in addition to cells, is predominantly constituted of hydrogels due to its biodegradability and ability to mimic the body’s anatomy and mechanical features, e.g. bones, etc. This review paper is devoted to explicating the bioprinting (3D/4D) of alginate hydrogels, which are extracts from brown algae, through extrusion additive manufacturing. Alginates are salt derivatives of alginic acid and constitute long chains of polysaccharides, which provides pliability and gelling adeptness to their structure. Alginate hydrogel (employed for extrusion) can be pristine or composite relying on the requisite properties (target application controlled or in vivo environment), e.g. alginate-natural (gelatin/agarose/collagen/hyaluronic acid/etc) and alginate-synthetic (polyethylene glycol (PEG)/pluronic F-127/etc). Extrusion additive manufacturing of alginate is preponderate among others with its uncomplicated processing, material efficiency (cut down on wastage), and outspread adaptability for viscosities (0.03–6 * 10 ⁴ Pa.s), but the procedure is limited by resolution (200 μ m) in addition to accuracy. However, 3D-fabricated biostructures display rigidness (unvarying with conditions) i.e. lacks a smart response, which is reassured by accounting time feature as a noteworthy accessory to printing, interpreted as 4D bioprinting. This review propounds the specific processing itinerary for alginate (meanwhile traversing across its composites/blends with natural and synthetic consideration) in extrusion along with its pre-/during/post-processing parameters intrinsic to the process. Furthermore, propensity is also presented in its (alginate extrusion processing) application for tissue engineering, i.e. bones, cartilage (joints), brain (neural), ear, heart (cardiac), eyes (corneal), etc, due to a worldwide quandary over accessibility to natural organs for diverse types of diseases. Additionally, the review contemplates recently invented advance printing, i.e. 4D printing for biotic species, with its challenges and future opportunities.