The lack of in vitro tissue and organ models capable of mimicking human physiology severely hinders the development and clinical translation of therapies and drugs with higher in vivo efficacy. Bioprinting allow us to fill this gap and generate 3D tissue analogues with complex functional and structural organization through the precise spatial positioning of multiple materials and cells. In this review, we report the latest developments in terms of bioprinting technologies for the manufacturing of cellular constructs with particular emphasis on material extrusion, jetting, and vat photopolymerization. We then describe the different base polymers employed in the formulation of bioinks for bioprinting and examine the strategies used to tailor their properties according to both processability and tissue maturation requirements. By relating function to organization in human development, we examine the potential of pluripotent stem cells in the context of bioprinting toward a new generation of tissue models for personalized medicine. We also highlight the most relevant attempts to engineer artificial models for the study of human organogenesis, disease, and drug screening. Finally, we discuss the most pressing challenges, opportunities, and future prospects in the field of bioprinting for tissue engineering (TE) and regenerative medicine (RM).