Bioprinting is an emerging technology with various applications in making functional tissue constructs to replace injured or diseased tissues. It is a relatively new approach that provides high reproducibility and precise control over the fabricated constructs in an automated manner, potentially enabling high-throughput production. During the bioprinting process, a solution of a biomaterial or a mixture of several biomaterials in the hydrogel form, usually encapsulating the desired cell types, termed the bioink, is used for creating tissue constructs. This bioink can be cross-linked or stabilized during or immediately after bioprinting to generate the final shape, structure, and architecture of the designed construct. Bioinks may be made from natural or synthetic biomaterials alone, or a combination of the two as hybrid materials. In certain cases, cell aggregates without any additional biomaterials can also be adopted for use as a bioink for bioprinting processes. An ideal bioink should possess proper mechanical, rheological, and biological properties of the target tissues, which are essential to ensure correct functionality of the bioprinted tissues and organs. In this review, we provide an in-depth discussion of the different bioinks currently employed for bioprinting, and outline some future perspectives in their further development.