Bioprinting aims to direct the spatial arrangement in three dimensions of cells, biomaterials, and growth factors. The biofabrication of clinically relevant constructs for the repair or modeling of either diseased or damaged tissues is rapidly advancing, resulting in the ability to three-dimensional (3D) print biomimetic platforms which imitate a large number of tissues in the human body. Primary tissue-specific cells are typically isolated from patients and used for the fabrication of 3D models for drug screening or tissue repair purposes. However, the lack of resilience of these platforms, due to the difficulties in harnessing, processing, and implanting patient-specific cells can limit regeneration ability. The printing of stem cells obviates these hurdles, producing functional in vitro models or implantable constructs. Advancements in biomaterial science are helping the development of inks suitable for the encapsulation and the printing of stem cells, promoting their functional growth and differentiation. This review specifically aims to investigate the most recent studies exploring innovative and functional approaches for the printing of 3D constructs to model disease or repair damaged tissues. Key concepts in tissue physiology are highlighted, reporting stem cell applications in biofabrication. Bioprinting technologies and biomaterial inks are listed and analyzed, including recent advancements in biomaterial design for bioprinting applications, commenting on the influence of biomaterial inks on the encapsulated stem cells. Ultimately, most recent successful efforts and clinical potentials for the manufacturing of functional physiological tissue substitutes are reported here, with a major focus on specific tissues, such as vasculature, heart, lung and airways, liver, bone and muscle.