The current need for organ and tissue replacement, repair and regeneration for patients is continually growing such that supply is not meeting the high demand primarily due to a paucity of donors as well as biocompatibility issues that lead to immune rejection of the transplant. In an effort to overcome these drawbacks, scientists working in the field of tissue engineering and regenerative medicine have investigated the use of scaffolds as an alternative to transplantation. These scaffolds are designed to mimic the extracellular matrix (ECM) by providing structural support as well as promoting attachment, proliferation, and differentiation with the ultimate goal of yielding functional tissues or organs. Initial attempts at developing scaffolds were problematic and subsequently inspired a growing interest in 3D printing as a mode for generating scaffolds. Utilizing three-dimensional printing (3DP) technologies, ECM-like scaffolds can be produced with a high degree of complexity and precision, where fine details can be included at a micron level. In this review, we discuss the criteria for printing viable and functional scaffolds, scaffolding materials, and 3DP technologies used to print scaffolds for tissue engineering. A hybrid approach, employing both natural and synthetic materials, as well as multiple printing processes may be the key to yielding an ECM-like scaffold with high mechanical strength, porosity, interconnectivity, biocompatibility, biodegradability, and high processability. Creating such biofunctional scaffolds could potentially help to meet the demand by patients for tissues and organs without having to wait or rely on donors for transplantation.