A current hypothesis gaining prominence proposes that activation of the enzyme telomerase is necessary for cells to become immortal, or capable of proliferating indefinitely. The theory suggests that almost all cancer cells must attain immortality for progression to malignant states and, hence, require activation of telomerase. This article reviews the function and formation of telomeres as background to evaluating the “telomere hypothesis.” Experiments in support of and experiments that challenge the hypothesis are examined. Possible approaches to telomerase inhibition are discussed. [J Natl Cancer Inst 87: 884–894, 1995]