We introduce new statistical methods for analyzing genomic datasets that measure many effects in many conditions (e.g. gene expression changes under many treatments). These new methods improve on existing methods by allowing for arbitrary correlations among conditions. This flexible approach increases power, improves effect-size estimates, and facilitates more quantitative assessments of effect-size heterogeneity than simple "shared/condition-specific" assessments. We illustrate these features through a detailed analysis of locally-acting ("cis") eQTLs in 44 human tissues (data from GTEx project). 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 highlights 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; transformed-fibroblasts). Our methods are widely applicable, computationally tractable for many conditions, and available at https://github.com/stephenslab/mashr.