The science of telomeres and telomerase has made tremendous progress in recent decades. In this review, we consider it first in a historical context (the Carrel–Hayflick–Olovnikov–Blackburn chain of discoveries) and then review current knowledge on the telomere structure and dynamics in norm and pathology. Central to the review are consequences of the telomere shortening, including telomere position effects, DNA damage signaling, and increased genetic instability. Cell senescence and role of telomere length in its development are discussed separately. Therapeutic aspects and risks of telomere lengthening methods including use of telomerase and other approaches are also discussed.