Analysis of cancer genomes has shown that a large fraction of chromosomal changes originate from catastrophic events including whole-genome duplication, chromothripsis, breakage-fusion-bridge cycles, and chromoplexy. Through sophisticated computational analysis of cancer genomes and experimental recapitulation of these catastrophic alterations, we have gained significant insights into the origin, mechanism, and evolutionary dynamics of cancer genome complexity. In this review, we summarize this progress and survey the major unresolved questions, with particular emphasis on the relative contributions of chromosome fragmentation and DNA replication errors to complex chromosomal alterations. Expected final online publication date for the Annual Review of Cancer Biology, Volume 6 is April 2022. Please see http://www.annualreviews.org/page/journal/pubdates for revised estimates.