We introduce new statistical methods for analyzing genomic datasets that measure many effects in many conditions (e.g. gene expression changes under many treat- ments). These new methods improve on existing methods by allowing for arbitrary correlations in effects among conditions. This flexible approach increases power, im- proves effect-size estimates, and facilitates more quantitative assessments of effect- size heterogeneity than simple “shared/condition-specific” assessments. We illustrate these features through three detailed analyses. The first is an assessment of locally- acting (“cis”) eQTLs in 44 human tissues. Our analysis identifies more eQTLs than existing approaches, consistent with improved power. More importantly, although eQTLs are often shared broadly among tissues, our more quantitative approach high- lights that effect sizes can vary considerably among tissues: some shared eQTLs show stronger effects in a subset of biologically-related tissues (e.g. brain-related tissues), or in only a single tissue (e.g. testis). We then apply our method to a setting in which all conditions are compared to a common control, as well as to an analysis considering the genetic effect on multiple diseases simultaneously. Our methods are widely applicable, computationally tractable for many conditions, and available at https://github.com/stephenslab/mashr.