Three-dimensional (3D) biofabrication techniques enable the production of multicellular tissue models as assay platforms for drug screening. The increased cellular and physiological complexity in these 3D tissue models should recapitulate the relevant biological environment found in the body. Here we describe the use of 3D bioprinting techniques to fabricate skin equivalent tissues of varying physiological complexity, including human epidermis, non-vascularized and vascularized full-thickness skin tissue equivalents, in a multi-well platform to enable drug screening. Human keratinocytes, fibroblasts, and pericytes, and induced pluripotent stem cell-derived endothelial cells were used in the biofabrication process to produce the varying complexity. The skin equivalents exhibit the correct structural markers of dermis and epidermis stratification, with physiological functions of the skin barrier. The robustness, versatility and reproducibility of the biofabrication techniques are further highlighted by the generation of atopic dermatitis (AD)-disease like tissues. These AD models demonstrate several clinical hallmarks of the disease, including: (i) spongiosis and hyperplasia; (ii) early and terminal expression of differentiation proteins; and (iii) increases in levels of pro-inflammatory cytokines. We show the pre-clinical relevance of the biofabricated AD tissue models to correct disease phenotype by testing the effects of dexamethasone, an anti-inflammatory corticosteroid, and three Janus Kinase inhibitors from clinical trials for AD. This study demonstrates the development of a versatile and reproducible bioprinting approach to create human skin equivalents with a range of cellular complexity for disease modeling. In addition, we establish several assay readouts that are quantifiable, robust, AD relevant, and can be scaled up for compound screening. The results show that the cellular complexity of the tissues develops a more physiologically relevant AD disease model. Thus, the skin models in this study offer an in vitro approach for the rapid understanding of pathological mechanisms, and testing for efficacy of action and toxic effects of drugs.