Engineered systems provide a means of deconstructing the microenvironmental cues which guide cell fate and function. These cues can include biochemical elements, as in ligand-receptor binding for cell-cell or cell-extracellular matrix (ECM) interactions, or responses to biophysical parameters, as in cell sensing of substrate material properties. One category of engineered system, high-throughput cell microarrays, are useful not only for the efficient identification of roles for new cues in specific biological processes but also for mapping combinatorial interactions between known cues. In his dissertation, Kerim describes a cell microarray platform with several additional capabilities: the integration of multiple readout modalities, including direct readout of mRNA expression using in situ hybridization and, crucially, cell-generated forces using traction force microscopy (TFM); and the deconvolution of signaling via cell-cell (ligand-receptor) interactions by combining cell-extrinsic ligand presentation with cell-intrinsic ligand knockdown. Kerim delineates the use of this platform towards investigations of microenvironmental regulation in the context of liver progenitor differentiation and lung tumor cell drug responses. Liver progenitor differentation was found to be combinatorially regulated by Notch, TGFβ signaling as well as interactions with ECM proteins. The Notch ligands Jag1 and D111 were further found to play distinct cell-intrinsic and cell-extrinsic roles in differentiation towards a biliary epithelial cell fate. Parallel TFM measurements in arrayed microenvironments indicated that progenitor cell differentiation towards biliary fates is a coordinated function of ECM composition, substrate stiffness, and cell contractility. Additional analysis of spatially-localized differentiation within array patterns showed that cooperative interactions between Notch and cell mechanotransduction signaling pathways are necessary for biliary differentiation. Similarly, the responses of tumor cells to drug treatment is known to depend on interactions with their matrix microenvironment. Lung tumor cell drug responses were mapped using a combinatorial ECM array design and shown to be a function not only of matrix composition but also genotype, specifically the presence or absence of the lineage oncogene ASCL1. Thus, this dissertation presents an advanced array platform which not only improves our understanding of biochemical and biophysical regulation of liver progenitor fate specification and lung tumor cell drug responses but also enables similar studies of other tissue contexts and organ systems.