To reduce expression of gene products not required under stress conditions, eukaryotic cells form large and complex cytoplasmic aggregates of RNA and proteins (stress granules; SGs), where transcripts are kept translationally inert. The overall composition of SGs, as well as their assembly requirements and regulation through stress-activated signaling pathways remain largely unknown. We have performed a genome-wide screen of S. cerevisiae gene deletion mutants for defects in SG formation upon glucose starvation stress. The screen revealed numerous genes not previously implicated in SG formation. Most mutants with strong phenotypes are equally SG defective when challenged with other stresses, but a considerable fraction is stress-specific. Proteins associated with SG defects are enriched in low-complexity regions, indicating that multiple weak macromolecule interactions are responsible for the structural integrity of SGs. Certain SG-defective mutants, but not all, display an enhanced heat-induced mutation rate. We found several mutations affecting the Ran GTPase, regulating nucleocytoplasmic transport of RNA and proteins, to confer SG defects. Unexpectedly, we found stress-regulated transcripts to reach more extreme levels in mutants unable to form SGs: stress-induced mRNAs accumulate to higher levels than in the wild-type, whereas stress-repressed mRNAs are reduced further in such mutants. Our findings are consistent with the view that, not only are SGs being regulated by stress signaling pathways, but SGs also modulate the extent of stress responses. We speculate that nucleocytoplasmic shuttling of RNA-binding proteins is required for gene expression regulation during stress, and that SGs modulate this traffic. The absence of SGs thus leads the cell to excessive, and potentially deleterious, reactions to stress.