Today, with the salient advancements of modern and smart technologies related to tissue engineering and regenerative medicine (TE-RM), the use of sustainable and biodegradable materials with biocompatibility and cost-effective advantages have been investigated more than before. Alginate as a naturally occurring anionic polymer can be obtained from brown seaweed to develop a wide variety of composites for TE, drug delivery, wound healing, and cancer therapy. This sustainable and renewable biomaterial displays several fascinating properties such as high biocompatibility, low toxicity, cost-effectiveness, and mild gelation by inserting divalent cations (e.g., Ca2+). In this context, challenges still exist in relation to the low solubility and high viscosity of high-molecular weight alginate, high density of intra- and inter-molecular hydrogen bonding, polyelectrolyte nature of the aqueous solution, and a lack of suitable organic solvents. Herein, TE-RM applications of alginate-based materials are deliberated, focusing on current trends, important challenges, and future prospects.