Setting time as the fourth dimension, 4D printing allows us to construct dynamic structures that can change their shape, property, or functionality over time under stimuli, leading to a wave of innovations in various fields. Recently, 4D printing of smart biomaterials, biological components, and living cells into dynamic living 3D constructs with 4D effects has led to an exciting field of 4D bioprinting. 4D bioprinting has gained increasing attention and is being applied to create programmed and dynamic cell-laden constructs such as bone, cartilage, and vasculature. This review presents an overview on 4D bioprinting for engineering dynamic tissues and organs, followed by a discussion on the approaches, bioprinting technologies, smart biomaterials and smart design, bioink requirements, and applications. While much progress has been achieved, 4D bioprinting as a complex process is facing challenges that need to be addressed by transdisciplinary strategies to unleash the full potential of this advanced biofabrication technology. Finally, we present future perspectives on the rapidly evolving field of 4D bioprinting, in view of its potential, increasingly important roles in the development of advanced dynamic tissues for basic research, pharmaceutics, and regenerative medicine.