Localised variation of somatic mutation rates affects diverse functional sequence elements in cancer genomes through poorly understood mutational processes. Here, we characterise the mutational landscape of 640,000 gene regulatory and chromatin architectural elements in 2,421 whole cancer genomes using our new statistical model RM2. This method quantifies differential mutation rates and signatures in classes of genomic elements via genetic, trinucleotide and megabase-scale effects. We report a detailed map of localised mutational processes affecting CTCF binding sites, transcription start sites (TSS) and cancer-specific open-chromatin regions. This includes a pan-cancer indel depletion in open-chromatin sites, a TSS-specific mutational process correlated with mRNA abundance in core cellular and cancer-associated processes, a subset of hypermutated, constitutively active CTCF binding sites involved in chromatin architectural interactions, and an enrichment of signature SBS17b in CTCF sites in gastrointestinal cancers. We also detect genetic driver alterations potentially underlying localised mutation rates, including RAD21 amplifications and BRAF mutations associating with mutagenesis of CTCF binding sites, and SDHA amplifications indicative of frequent lung cancer mutations in open-chromatin sites. Our framework and the catalogue of localised mutational processes provide novel perspectives to cancer genome evolution and its implications for oncogenesis, tumor heterogeneity and cancer driver gene discovery.