Cancer is a disease of the genome, but the dramatic inter-patient variability in mutation number is poorly understood. Tumours of the same type can differ by orders of magnitude in their mutation rate. To understand potential drivers and consequences of the underlying heterogeneity in mutation rate across tumours, we evaluated both local and global measures of mutation density: both single-stranded and double-stranded DNA breaks in 2,460 tumours of 38 cancer types. We find that SCNAs in thousands of genes are associated with elevated rates of point-mutations, while similarly point-mutation patterns in dozens of genes are associated with specific patterns of DNA double-stranded breaks. These candidate drivers of mutation density are enriched for known cancer drivers, and preferentially occur early in tumour evolution, appearing clonally in all cells of a tumour. To supplement this understanding of global mutation density, we developed and validated a tool called SeqKat to identify localized "rainstorms" of point-mutations (kataegis). We show that rates of kataegis differ by four orders of magnitude across tumour types, with malignant lymphomas showing the highest. Tumours with TP53 mutations were 2.6-times more likely to harbour a kataegic event than those without, and 239 SCNAs were associated with elevated rates of kataegis, including loss of the tumour-suppressor CDKN2A. We identify novel subtypes of kataegic events not associated with aberrant APOBEC activity, and find that these are localized to specific cellular regions, enriched for MYC-target genes. Kataegic events were associated with patient survival in some, but not all tumour types, highlighting a combination of global and tumour-type specific effects. Taken together, we reveal a landscape of genes driving localized and tumour-specific hyper-mutation, and reveal novel mutational processes at play in specific tumour types.