We introduce new statistical methods for analyzing genomic data sets that measure many effects in many conditions (for example, gene expression changes under many treatments). These new methods improve on existing methods by allowing for arbitrary correlations in effect sizes among conditions. This flexible approach increases power, improves effect estimates and allows for more quantitative assessments of effect-size heterogeneity compared to simple shared or condition-specific assessments. We illustrate these features through an analysis of locally acting variants associated with gene expression (cis expression quantitative trait loci (eQTLs)) in 44 human tissues. Our analysis identifies more eQTLs than existing approaches, consistent with improved power. We show that although genetic effects on expression are extensively shared among tissues, effect sizes can still vary greatly among tissues. Some shared eQTLs show stronger effects in subsets of biologically related tissues (for example, brain-related tissues), or in only one tissue (for example, testis). Our methods are widely applicable, computationally tractable for many conditions and available online.