Clinical and laboratory studies over recent decades have established branched evolution as a feature of cancer. However, while grounded in somatic selection, several lines of evidence suggest a Darwinian model alone is insufficient to fully explain cancer evolution. First, the role of macroevolutionary events in tumour initiation and progression contradicts Darwin's central thesis of gradualism. Whole-genome doubling, chromosomal chromoplexy and chromothripsis represent examples of single catastrophic events which can drive tumour evolution. Second, neutral evolution can play a role in some tumours, indicating that selection is not always driving evolution. Third, increasing appreciation of the role of the ageing soma has led to recent generalised theories of age-dependent carcinogenesis. Here, we review these concepts and others, which collectively argue for a model of cancer evolution which extends beyond Darwin. We also highlight clinical opportunities which can be grasped through targeting cancer vulnerabilities arising from non-Darwinian patterns of evolution.