Numerical chromosomal instability is a ubiquitous feature of human neoplasms. Due to experimental limitations, fundamental characteristics of karyotypic changes in cancer are poorly understood. Using an experimentally inspired stochastic model, based on the potency and chromosomal distribution of oncogenes and tumor suppressor genes, we show that cancer cells have evolved to exist within a narrow range of chromosome missegregation rates that optimizes phenotypic heterogeneity and clonal survival. Departure from this range reduces clonal fitness and limits subclonal diversity. Mapping of the aneuploid fitness landscape reveals a highly favorable, commonly observed, near-triploid state onto which evolving diploid- and tetraploid-derived populations spontaneously converge, albeit at a much lower fitness cost for the latter. Finally, by analyzing 1,368 chromosomal translocation events in five human cancers, we find that karyotypic evolution also shapes chromosomal translocation patterns by selecting for more oncogenic derivative chromosomes. Thus, chromosomal instability can generate the heterogeneity required for Darwinian tumor evolution.