Telomere maintenance is essential for maintaining genome integrity in both normal and cancer cells. Without functional telomeres, chromosomes lose their protective structure and undergo fusion and breakage events that drive further genome instability, including cell arrest or death. One means by which this loss can be overcome in stem cells and cancer cells is via re‐addition of G‐rich telomeric repeats by the telomerase reverse transcriptase (TERT). During aging of somatic tissues, however, insufficient telomerase expression leads to a proliferative arrest called replicative senescence, which is triggered when telomeres reach a critically short threshold that induces a DNA damage response. Cancer cells express telomerase but do not entirely escape telomere instability as they often possess short telomeres; hence there is often selection for genetic alterations in the TERT promoter that result in increased telomerase expression. In this review, we discuss our current understanding of the consequences of telomere instability in cancer and aging, and outline the opportunities and challenges that lie ahead in exploiting the reliance of cells on telomere maintenance for preserving genome stability.