Regional mutagenesis in cancer genomes associates with DNA replication timing (RT) and chromatin accessibility (CA) of normal cells, however human cancer epigenomes remain uncharacterized in this context. Here we model megabase-scale mutation frequencies in 2517 cancer genomes with 773 CA and RT profiles of cancers and normal cells. We find that CA profiles of matching cancers, rather than normal cells, predict regional mutagenesis and mutational signatures, indicating that most passenger mutations follow the epigenetic landscapes of transformed cells. Carcinogen-induced and unannotated signatures show the strongest associations with epigenomes. Associations with normal cells in melanomas, lymphomas and SBS1 signatures suggest earlier occurrence of mutations in cancer evolution. Frequently mutated regions unexplained by CA and RT are enriched in cancer genes and developmental pathways, reflecting contributions of localized mutagenesis and positive selection. These results underline the complex interplay of mutational processes, genome function and evolution in cancer and tissues of origin.