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Expansions and contractions of tandem DNA repeats generate genetic variation in human populations and in human tissues. Some expanded repeats cause inherited disorders and some are also somatically unstable. Here we analysed DNA sequencing data from over 900,000 participants in the UK Biobank and the All of Us Research Program using computational approaches to recognize, measure and learn from DNA-repeat instability. Repeats at different loci exhibited widely variable tissue-specific propensities to mutate in the germline and blood. Common alleles of repeats in TCF4 and ADGRE2 exhibited high rates of length mosaicism in the blood, demonstrating that most human genomes contain repeat elements that expand as we age. Genome-wide association analyses of the extent of somatic expansion of unstable repeat alleles identified 29 loci at which inherited variants increased expansion of one or more DNA repeats in blood (P = 5 × 10 to 2.5 × 10). These genetic modifiers exhibited strong collective effects on repeat instability: at one repeat, somatic expansion rates varied fourfold between individuals with the highest and lowest 5% of polygenic scores. Modifier alleles at several DNA-repair genes exhibited opposite effects on the blood instability of the TCF4 repeat compared with other DNA repeats. Expanded repeats in the 5' untranslated region of the glutaminase (GLS) gene associated with stage 5 chronic kidney disease (odds ratio (OR) = 14.0 (5.7-34.3, 95% confidence interval (CI))) and liver diseases (OR = 3.0 (1.5-5.9, 95% CI)). These results point to complex dynamics of DNA repeats in human populations and across the human lifespan.