Tandemly repeated sequences are highly mutable and variable features of genomes. Tandem repeat expansions are responsible for a growing list of human diseases, even though it is hard to determine tandem repeat sequences with current DNA sequencing technology. Recent long-read technologies are promising, because the DNA reads are often longer than the repetitive regions, but are hampered by high error rates. Here, we report robust detection of human repeat expansions from careful alignments of long (PacBio and nanopore) reads to a reference genome. Our method (tandem-genotypes) is robust to systematic sequencing errors, inexact repeats with fuzzy boundaries, and low sequencing coverage. By comparing to healthy controls, we can prioritize pathological expansions within the top 10 out of 700000 tandem repeats in the genome. This may help to elucidate the many genetic diseases whose causes remain unknown.