Three-dimensional (3D) bioprinting is a promising and rapidly evolving technology in the field of additive manufacturing. It enables the fabrication of living cellular constructs with complex architectures that are suitable for various biomedical applications, such as tissue engineering, disease modeling, drug screening, and precision regenerative medicine. The ultimate goal of bioprinting is to produce stable, anatomically-shaped, human-scale functional organs or tissue substitutes that can be implanted. Although various bioprinting techniques have emerged to develop customized tissue-engineering substitutes over the past decade, several challenges remain in fabricating volumetric tissue constructs with complex shapes and sizes and translating the printed products into clinical practice. Thus, it is crucial to develop a successful strategy for translating research outputs into clinical practice to address the current organ and tissue crises and improve patients' quality of life. This review article discusses the challenges of the existing bioprinting processes in preparing clinically relevant tissue substitutes. It further reviews various strategies and technical feasibility to overcome the challenges that limit the fabrication of volumetric biological constructs and their translational implications. Additionally, the article highlights exciting technological advances in the 3D bioprinting of anatomically shaped tissue substitutes and suggests future research and development directions. This review aims to provide readers with insight into the state-of-the-art 3D bioprinting techniques as powerful tools in engineering functional tissues and organs.