Three-dimensional (3D) bioprinting technology has emerged as a powerful biofabrication platform for tissue engineering because of its ability to engineer living cells and biomaterial-based 3D objects. Over the last few decades, droplet-based, extrusion-based, and laser-assisted bioprinters have been developed to fulfill certain requirements in terms of resolution, cell viability, cell density, etc. Simultaneously, various bio-inks based on natural–synthetic biomaterials have been developed and applied for successful tissue regeneration. To engineer more realistic artificial tissues/organs, mixtures of bio-inks with various recipes have also been developed. Taken together, this review describes the fundamental characteristics of the existing bioprinters and bio-inks that have been currently developed, followed by their advantages and disadvantages. Finally, various tissue engineering applications using 3D bioprinting are briefly introduced.