Aneuploidy is a very rare and tissue-specific event in normal conditions, occurring in a low number of brain and liver cells. Its frequency increases in age-related disorders and is one of the hallmarks of cancer. Aneuploidy has been associated with defects in the spindle assembly checkpoint (SAC). However, the relationship between chromosome number alterations, SAC genes and tumor susceptibility remains unclear. Here, we provide a comprehensive review of SAC gene alterations at genomic and transcriptional level across human cancers and discuss the oncogenic and tumor suppressor functions of aneuploidy. SAC genes are rarely mutated but frequently overexpressed, with a negative prognostic impact on different tumor types. Both increased and decreased SAC gene expression show oncogenic potential in mice. SAC gene upregulation may drive aneuploidization and tumorigenesis through mitotic delay, coupled with additional oncogenic functions outside mitosis. The genomic background and environmental conditions influence the fate of aneuploid cells. Aneuploidy reduces cellular fitness. It induces growth and contact inhibition, mitotic and proteotoxic stress, cell senescence and production of reactive oxygen species. However, aneuploidy confers an evolutionary flexibility by favoring genome and chromosome instability (CIN), cellular adaptation, stem cell-like properties and immune escape. These properties represent the driving force of aneuploid cancers, especially under conditions of stress and pharmacological pressure, and are currently under investigation as potential therapeutic targets. Indeed, promising results have been obtained from synthetic lethal combinations exploiting CIN, mitotic defects, and aneuploidy-tolerating mechanisms as cancer vulnerability.