Novel additive manufacturing techniques are revolutionizing fields of industry providing more dimensions to control and the versatility of fabricating multi-material products. Medical applications hold great promise to manufacture constructs of mixed biologically compatible materials together with functional cells and tissues. We reviewed technologies and promising developments nurturing innovation of physiologically relevant models to study safety of chemicals that are hard to reproduce in current models, or diseases for which there are no models available. Extrusion-, inkjet- and laser-assisted bioprinting are the most used techniques. Hydrogels as constituents of bioinks and biomaterial inks are the most versatile materials to recreate physiological and pathophysiological microenvironments. The highlighted bioprinted models were chosen because they guarantee post-printing cellular viability while maintaining desirable mechanical properties of their constitutive bioinks or biomaterial inks to ensure their printability. Bioprinting is being readily adopted to overcome ethical concerns of in vivo models and improve the automation, reproducibility, geometry stability of traditional in vitro models. The challenges for advancing the technological level readiness of bioprinting require overcoming heterogeneity, microstructural complexity, dynamism and integration with other models, to generate multi-organ platforms that can inform about biological responses to chemical exposure, disease development and efficacy of novel therapies.