It has been observed that bone fractures carry a risk of high mortality and morbidity. The deployment of a proper bone healing method is essential to achieve the desired success. Over the years, bone tissue engineering (BTE) has appeared to be a very promising approach aimed at restoring bone defects. The main role of the BTE is to apply new, efficient, and functional bone regeneration therapy via a combination of bone scaffolds with cells and/or healing promotive factors (e.g., growth factors and bioactive agents). The modern approach involves also the production of living bone grafts in vitro by long-term culture of cell-seeded biomaterials, often with the use of bioreactors. This review presents the most recent findings concerning biomaterials, cells, and techniques used for the production of living bone grafts under in vitro conditions. Particular attention has been given to features of known bioreactor systems currently used in BTE: perfusion bioreactors, rotating bioreactors, and spinner flask bioreactors. Although bioreactor systems are still characterized by some limitations, they are excellent platforms to form bioengineered living bone grafts in vitro for bone fracture regeneration. Moreover, the review article also describes the types of biomaterials and sources of cells that can be used in BTE as well as the role of three-dimensional bioprinting and pulsed electromagnetic fields in both bone healing and BTE.