Cancer genome sequencing provides the first direct information on how mutation rates vary across the human genome in somatic cells. Testing diverse genetic and epigenetic features, here we show that mutation rates in cancer genomes are strikingly related to chromatin organization. Indeed, at the megabase scale, a single feature--levels of the heterochromatin-associated histone modification H3K9me3--can account for more than 40% of mutation-rate variation, and a combination of features can account for more than 55%. The strong association between mutation rates and chromatin organization is upheld in samples from different tissues and for different mutation types. This suggests that the arrangement of the genome into heterochromatin- and euchromatin-like domains is a dominant influence on regional mutation-rate variation in human somatic cells.