Nanotechnology is a rapidly emerging technology dealing with so-called nanomaterials which at least in one dimension have size smaller than 100 nm. One of the most potentially promising applications of nanotechnology is in the area of tissue engineering, including biofabrication of 3D human tissues and organs. This paper focused on demonstrating how nanomaterials with nanolevel size can contribute to development of 3D human tissues and organs which have macrolevel organization. Specific nanomaterials such as nanofibers and nanoparticles are discussed in the context of their application for biofabricating 3D human tissues and organs. Several examples of novel tissue and organ biofabrication technologies based on using novel nanomaterials are presented and their recent limitations are analyzed. A robotic device for fabrication of compliant composite electrospun vascular graft is described. The concept of self-assembling magnetic tissue spheroids as an intermediate structure between nano- and macrolevel organization and building blocks for biofabrication of complex 3D human tissues and organs is introduced. The design of in vivo robotic bioprinter based on this concept and magnetic levitation of tissue spheroids labeled with magnetic nanoparticles is presented. The challenges and future prospects of applying nanomaterials and nanotechnological strategies in organ biofabrication are outlined.