Conventional approaches in drug development involve testing on 2D-cultured mammalian cells, followed by experiments in rodents. Although this is the common strategy, it has significant drawbacks: in 2D cell culture with human cells, the cultivation at normoxic conditions on a plastic or glass surface is an artificial situation that significantly changes energy metabolism, shape and intracellular signaling, which in turn directly affects drug response. On the other hand, rodents as the most frequently used animal models have evolutionarily separated from primates about 100 million years ago, with significant differences in physiology, which frequently leads to results not reproducible in humans. As an alternative, spheroid technology and micro-organoids have evolved in the last decade to provide 3D context for cells similar to native tissue. However, organoids used for drug testing are usually just in the 50–100 micrometers range and thereby too small to mimic micro-environmental tissue conditions such as limited nutrient and oxygen availability. An attractive alternative offers 3D bioprinting as this allows fabrication of human tissue equivalents from scratch with hollow structures for perfusion and strict spatiotemporal control over the deposition of cells and extracellular matrix proteins. Thereby, tissue surrogates with defined geometry are fabricated that offer unique opportunities in exploring cellular cross-talk, mechanobiology and morphogenesis. These tissue-equivalents are also very attractive tools in drug testing, as bioprinting enables standardized production, parallelization, and application-tailored design of human tissue, of human disease models and patient-specific tissue avatars. This review, therefore, summarizes recent advances in 3D bioprinting technology and its application for drug screening.