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In the evolution of species, the karyotype changes with a timescale of tens to hundreds of thousand years. In the development of cancer, the karyotype often is modified in cancerous cells over the lifetime of an individual. Characterizing these changes and understanding the mechanisms leading to them has been of interest in a broad range of disciplines including evolution, cytogenetics, and cancer genetics. A central issue relates to the relative roles of random vs deterministic mechanisms in shaping the changes. Although it is possible that all changes result from random events followed by selection, many results point to other non-random factors that play a role in karyotype evolution. In cancer, chromosomal instability leads to characteristic changes in the karyotype, in which different individuals with a specific type of cancer display similar changes in karyotype structure over time. Statistical analyses of chromosome lengths in different species indicate that the length distribution of chromosomes is not consistent with models in which the lengths of chromosomes are random or evolve solely by simple random processes. A better understanding of the mechanisms underlying karyotype evolution should enable the development of quantitative theoretical models that combine the random and deterministic processes that can be compared to experimental determinations of the karyotype in diverse settings.