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Identifying the causal variants and mechanisms that drive complex traits and diseases remains a core problem in human genetics. Most of these variants individually have weak effects and lie in non-coding gene-regulatory elements, for which we lack a complete understanding of how single-nucleotide alterations modulate transcriptional processes to affect human phenotypes. To address this problem, we measured the activity of 221,412 fine-mapped trait-associated variants using a massively parallel reporter assay in 5 diverse cell types. We show that this assay effectively discriminates between likely causal variants and controls, and identified 13,121 regulatory variants with high precision. Although the effects of these variants largely agree with orthogonal measures of function, only 69% of them can plausibly be explained by the disruption of a known transcription factor binding motif. We investigated the mechanisms of 136 variants using saturation mutagenesis and assigned affected transcription factors for 91% of variants without a clear canonical mechanism. Finally, we detected regulatory epistasis at 11% of tested regulatory variants in close proximity and identified multiple functional variants on the same haplotype at a small, but important, subset of trait-associated loci. Overall, our study provides a systematic functional characterization of likely causal common variants that underlie complex and molecular human traits, enabling new insights into the regulatory grammar underlying disease risk.