Remarkable advancement in 3D printing technology in recent years has already transformed many aspects of industrial manufacturing. The immense potential of 3D printing is already being explored in state-of-the-art biomedical research field. Often termed "bioprinting", 3D printing is utilized to generate biological structures with high resolution and specificity for tissue engineering and regenerative medical applications. With the maturation of bioprinting apparatus, now the focus is shifting to engineering "bioinks" that can accommodate the versatility of biological systems, while still maintaining their printability. In this review, bioink technologies based on various polymers to produce soft biomaterials, such as hydrogels and elastomers, having a diverse array of physicochemical and bioactive properties are introduced and highlighted.