Most genetic association studies focus on binary variants. To identify the effects of multi-allelic variation of tandem repeats (TRs) on human traits, we performed direct TR genotyping and phenome-wide association studies in 168,554 individuals from the UK Biobank, identifying 47 TRs showing causal associations with 73 traits. We replicated 23 of 31 (74%) of these causal associations in the All of Us cohort. While this set included several known repeat expansion disorders, novel associations we found were attributable to common polymorphic variation in TR length rather than rare expansions and include e.g. a coding polyhistidine motif in HRCT1 influencing risk of hypertension and a poly(CGC) in the 5'UTR of GNB2 influencing heart rate. Causal TRs were strongly enriched for associations with local gene expression and DNA methylation. Our study highlights the contribution of multi-allelic TRs to the "missing heritability" of the human genome.