The review considers the function of the important chromosome regions telomeres in normal and immortal cells. Telomeres are dynamic nucleoprotein structures that cap the ends of eukaryotic chromosomes, protecting them from degradation and end-to-end fusion. The functional state of telomeres depends on many interrelated parameters such as telomerase activity, the status of the telomere safety complex shelterin, and telomere-associated proteins (replication, recombination, DNA break repair factors, etc.). Special attention is paid to the mechanisms that control the telomere length in normal and immortal cells as well as in cells containing or lacking active telomerase. The features attributed to an alternative telomere length control are analyzed, in particular, in view of a recently discovered additional mechanism of telomere shortening by t-cycle trimming. The possibility of expressing both telomerase-dependent and recombinational pathways of telomere length control in normal mammalian cells is considered, as well as the role of shelterin proteins in choosing one of them to be dominant. The review additionally discusses the role of telomeres in the spatial organization of the nucleus during mitosis and meiosis and specific telomere organizations in mammals, including Iberian shrews with their unusual or rare chromosome structures.