Bioprinting aims to provide new avenues for regenerating damaged human tissues through the controlled printing of live cells and biocompatible materials that can function therapeutically. Polymeric hydrogels are commonly investigated ink materials for 3D and 4D bioprinting applications, as they can contain intrinsic properties relative to those of the native tissue extracellular matrix and can be printed to produce scaffolds of hierarchical organization. The incorporation of nanoscale material additives, such as nanoparticles, to the bulk of inks, has allowed for significant tunability of the mechanical, biological, structural, and physicochemical material properties during and after printing. The modulatory and biological effects of nanoparticles as bioink additives can derive from their shape, size, surface chemistry, concentration, and/or material source, making many configurations of nanoparticle additives of high interest to be thoroughly investigated for the improved design of bioactive tissue engineering constructs. This paper aims to review the incorporation of nanoparticles, as well as other nanoscale additive materials, to printable bioinks for tissue engineering applications, specifically bone, cartilage, dental, and cardiovascular tissues. An overview of the various bioinks and their classifications will be discussed with emphasis on cellular and mechanical material interactions, as well the various bioink formulation methodologies for 3D and 4D bioprinting techniques. The current advances and limitations within the field will be highlighted.