Tumor genomes are generally thought to evolve through a gradual accumulation of mutations, but the observation that extraordinarily complex rearrangements can arise through single mutational events suggests that evolution may be accelerated by punctuated changes in genome architecture. To assess the prevalence and origins of complex genomic rearrangements (CGRs), we mapped 6179 somatic structural variation breakpoints in 64 cancer genomes from seven tumor types and screened for clusters of three or more interconnected breakpoints. We find that complex breakpoint clusters are extremely common: 154 clusters comprise 25% of all somatic breakpoints, and 75% of tumors exhibit at least one complex cluster. Based on copy number state profiling, 63% of breakpoint clusters are consistent with being CGRs that arose through a single mutational event. CGRs have diverse architectures including focal breakpoint clusters, large-scale rearrangements joining clusters from one or more chromosomes, and staggeringly complex chromothripsis events. Notably, chromothripsis has a significantly higher incidence in glioblastoma samples (39%) relative to other tumor types (9%). Chromothripsis breakpoints also show significantly elevated intra-tumor allele frequencies relative to simple SVs, which indicates that they arise early during tumorigenesis or confer selective advantage. Finally, assembly and analysis of 4002 somatic and 6982 germline breakpoint sequences reveal that somatic breakpoints show significantly less micro-homology and fewer templated insertions than germline breakpoints, and this effect is stronger at CGRs than at simple variants. These results are inconsistent with replication-based models of CGR genesis and strongly argue that non-homologous repair of concurrently arising DNA double-strand breaks is the predominant mechanism underlying complex cancer genome rearrangements.