3D biomaterial printing has emerged as a potentially revolutionary technology, promising to transform both research and medical therapeutics. Although there has been recent progress in the field, on-demand fabrication of functional and transplantable tissues and organs is still a distant reality. To advance to this point, there are two major technical challenges that must be overcome. The first is expanding upon the limited variety of available 3D printable biomaterials (biomaterial inks), which currently do not adequately represent the physical, chemical, and biological complexity and diversity of tissues and organs within the human body. Newly developed biomaterial inks and the resulting 3D printed constructs must meet numerous interdependent requirements, including those that lead to optimal printing, structural, and biological outcomes. The second challenge is developing and implementing comprehensive biomaterial ink and printed structure characterization combined with in vitro and in vivo tissue- and organ-specific evaluation. This perspective outlines considerations for addressing these technical hurdles that, once overcome, will facilitate rapid advancement of 3D biomaterial printing as an indispensable tool for both investigating complex tissue and organ morphogenesis and for developing functional devices for a variety of diagnostic and regenerative medicine applications.