Treating large bone defects, known as critical-sized defects (CSDs), is challenging because they are not spontaneously healed by the patient’s body. Due to the limitations associated with conventional bone grafts, bone tissue engineering (BTE), based on three-dimensional (3D) bioprinted scaffolds, has emerged as a promising approach for bone reconstitution and treatment. Bioprinting technology allows for incorporation of living cells and/or growth factors into scaffolds aiming to mimic the structure and properties of the native bone. To date, a wide range of biomaterials (either natural or synthetic polymers), as well as various cells and growth factors, have been explored for use in scaffold bioprinting. However, a key challenge that remains is the fabrication of scaffolds that meet structure, mechanical, and osteoconductive requirements of native bone and support vascularization. In this review, we briefly present the latest developments and discoveries of CSD treatment by means of bioprinted scaffolds, with a focus on the biomaterials, cells, and growth factors for formulating bioinks and their bioprinting techniques. Promising state-of-the-art pathways or strategies recently developed for bioprinting bone scaffolds are highlighted, including the incorporation of bioactive ceramics to create composite scaffolds, the use of advanced bioprinting technologies (e.g., core/shell bioprinting) to form hybrid scaffolds or systems, as well as the rigorous design of scaffolds by taking into account of the influence of such parameters as scaffold pore geometry and porosity. We also review in-vitro assays and in-vivo models to track bone regeneration, followed by a discussion of current limitations associated with 3D bioprinting technologies for BTE. We conclude this review with emerging approaches in this field, including the development of gradient scaffolds, four-dimensional (4D) printing technology via smart materials, organoids, and cell aggregates/spheroids along with future avenues for related BTE.