Somatic mutations are an inevitable component of ageing and the most important cause of cancer. The rates and types of somatic mutation vary across individuals, but relatively few inherited influences on mutation processes are known. We performed a comprehensive gene-based rare variant association study with diverse mutational processes, using human cancer genomes from over 11,000 individuals of European ancestry. By combining burden and variance tests, we identify 207 associations involving 15 somatic mutational phenotypes and 42 genes that replicated in an independent data set at a FDR of 1%. We associated rare inherited deleterious variants in novel genes such as MSH3, EXO1, SETD2, and MTOR with two different forms of DNA mismatch repair deficiency, and variants in genes such as EXO1, PAXIP1, and WRN with deficiency in homologous recombination repair. In addition, we identified associations with other mutational processes, such as APEX1 with APOBEC-signature mutagenesis. Many of the novel genes interact with each other and with known mutator genes within cellular sub-networks. Considered collectively, damaging variants in the newly-identified genes are prevalent in the population. We suggest that rare germline variation in diverse genes commonly impacts mutational processes in somatic cells.