A key mutational process in cancer is structural variation, in which rearrangements delete, amplify or reorder genomic segments that range in size from kilobases to whole chromosomes1-7. Here we develop methods to group, classify and describe somatic structural variants, using data from the Pan-Cancer Analysis of Whole Genomes (PCAWG) Consortium of the International Cancer Genome Consortium (ICGC) and The Cancer Genome Atlas (TCGA), which aggregated whole-genome sequencing data from 2,658 cancers across 38 tumour types8. Sixteen signatures of structural variation emerged. Deletions have a multimodal size distribution, assort unevenly across tumour types and patients, are enriched in late-replicating regions and correlate with inversions. Tandem duplications also have a multimodal size distribution, but are enriched in early-replicating regions-as are unbalanced translocations. Replication-based mechanisms of rearrangement generate varied chromosomal structures with low-level copy-number gains and frequent inverted rearrangements. One prominent structure consists of 2-7 templates copied from distinct regions of the genome strung together within one locus. Such cycles of templated insertions correlate with tandem duplications, and-in liver cancer-frequently activate the telomerase gene TERT. A wide variety of rearrangement processes are active in cancer, which generate complex configurations of the genome upon which selection can act.