The replication of linear chromosome DNA by DNA polymerase leads to the loss of terminal sequences, in the absence of a special mechanism to maintain ends or telomeres. This mechanism is known to consist of short terminal repeats and the enzyme telomerase, which contains RNA complementary to the DNA repeats. There is evidence that telomeric DNA continually decreases in size in the absence of telomerase, and this is followed by cellular senescence. Immortalisation of somatic cells is accompanied, at least in some cases, by acquisition of telomerase activity. The cloning of DNA coding for the RNA component of telomerase has opened up some new experimental approaches, including the study of telomerases with mutant RNA. The telomere theory of cellular senescence appears to provide a molecular basis for the 'Hayflick limit' to human fibroblast growth. However the telomeres and behaviour of primary mouse cells are anomolous, and many immortalised human cell lines lack normal telomerase activity. These exceptions are not easily accommodated in the telomere theory.