Escalating bone graft scarcity and donor site morbidity worldwide are alarming reminders that highlight the need for alternatives to gold standard tissue rejuvenation methods. Over the last few decades, many efforts have been made in bone tissue engineering (BTE) to fabricate artificial bone transplants. Conventional BTE techniques do not render pertinent spatial organization of cells, and they fail in mimicking the extracellular matrix of native bone tissue. This setback can be overcome by using the emerging technology of three-dimensional bioprinting (3DBP). 3DBP is a state-of-the-art technology that provides accurate hierarchal biomaterial structures that accommodate live-cell patterning to mimic their native counterparts. Herein, we provide an overview on the recent progress of cell-laden 3DBP technologies and also discuss the various biomaterials utilized (natural polymers such as chitosan, collagen, gelatin, hyaluronic acid, and silk fibroin and synthetic polymers such as PCL, PVP, and ceramics) to engineer scaffolds with requisite structural, mechanical, and biological complexity. We also highlight some of the persisting challenges and the solutions to surmount them, paving the way for progress in the field. Finally, we discuss how the combination of novel modalities with 3DBP can pave the way for new frontiers, like four-dimensional bioprinting (4DBP), to bring customized, stimuli-responsive, and highly effective regenerative scaffolds to bone tissue engineering.