Telomeres are the specialized DNA-protein structures that cap the ends of linear chromosomes, thereby protecting them from degradation and fusion by cellular DNA repair processes. In vertebrate cells, telomeres consist of several kilobase pairs of DNA having the sequence TTAGGG, a few hundred base pairs of single-stranded DNA at the 3' end of the telomeric DNA tract, and a host of proteins that organize the telomeric double and single-stranded DNA into a protective structure. Functional telomeres are essential for maintaining the integrity and stability of genomes. When combined with loss of cell cycle checkpoint controls, telomere dysfunction can lead to genomic instability, a common cause and hallmark of cancer. Consequently, normal mammalian cells respond to dysfunctional telomeres by undergoing apoptosis (programmed cell death) or cellular senescence (permanent cell cycle arrest), two cellular tumor suppressor mechanisms. These tumor suppressor mechanisms are potent suppressors of cancer, but recent evidence suggests that they can antagonistically also contribute to aging phenotypes. Here, we review what is known about the structure and function of telomeres in mammalian cells, particularly human cells, and how telomere dysfunction may arise and contribute to cancer and aging phenotypes.