Three-dimensional (3D) printing, which is a valuable technique for the fabrication of tissue-engineered constructs and biomedical devices with complex architectures, has brought about considerable progress in regenerative medicine, drug delivery, and diagnosis of diseases. However, because of the static and inanimate properties of conventional 3D-printed structures, it is difficult to use them in therapies for active and precise medicine, such as improved tissue regeneration, targeted or controlled drug delivery, and advanced pathophysiological monitoring. The integration of stimuli-responsive biomaterials into 3D printing provides a potential strategy for designing and building smart constructs that exhibit programmed functions and controllable changes in properties in response to exogenous and autogenous stimuli. These features make 3D-printed smart constructs the next generation of tissue-engineered products. In this review, we introduce the prevalent 3D printing techniques (with an emphasis on the differences between 3D printing and bioprinting, and biomaterials and bioink), the working principle of each technique, and the advantages of using 3D printing for the fabrication of smart constructs. Stimuli-responsive biomaterials that are widely used for 3D printing of smart constructs are categorized, followed by a summary of their applications in tissue regeneration, drug delivery, and biosensors. Finally, the challenges and future perspectives of 3D-printed smart constructs are discussed.