Biofabrication techniques allow for the construction of biocompatible and biofunctional structures composed from biomaterials, cells and biomolecules. Bioprinting is an emerging 3D printing method which utilizes biomaterial-based mixtures with cells and other biological constituents into printable suspensions known as bioinks. Coupled with automated design protocols and based on different modes for droplet deposition, 3D bioprinters are able to fabricate hydrogel-based objects with specific architecture and geometrical properties, providing the necessary environment that promotes cell growth and directs cell differentiation towards application-related lineages. For the preparation of such bioinks, various water-soluble biomaterials have been employed, including natural and synthetic biopolymers, and inorganic materials. Bioprinted constructs are considered to be one of the most promising avenues in regenerative medicine due to their native organ biomimicry. For successful application, the bioprinted constructs should meet particular criteria such as optimal biological response, mechanical properties similar to the target tissue, high levels of reproducibility and printing fidelity, but also increased upscaling capability. In this review, we highlight the most recent advances in bioprinting, focusing on the regeneration of various tissues including bone, cartilage, cardiovascular, neural, skin, and other organs such as liver, kidney, pancreas, and lungs. We discuss the rapidly developing co-culture bioprinting systems used to resemble the complexity of tissues and organs and the crosstalk between various cell populations towards regeneration. Moreover, we report on the basic physical principles governing 3D bioprinting, and the ideal bioink properties based on the biomaterials’ regenerative potential. We examine and critically discuss the present status of 3D bioprinting regarding its applicability and current limitations that need to be overcome to establish it at the forefront of artificial organ production and transplantation.