Telomeres protect chromosome ends from being detected as lesions and from triggering DNA damage checkpoints. Paradoxically, telomere function depends on checkpoint proteins such as ATM and ATR, but a molecular model explaining this seemingly contradictory relationship has been missing so far. Here we show that the DNA damage machinery acts on telomeres in at least two independent steps. First, the ATR-dependent machinery is recruited to telomeres before telomere replication is completed, likely in response to single-stranded DNA resulting from replication fork stalling. Second, after replication, telomeres attract ATM and the homologous recombination (HR) machinery. In vivo and in vitro results suggest that the HR machinery is required for formation of a telomere-specific structure at chromosome ends after replication. Our results suggest that telomere ends need to be recognized as DNA damage to complete end replication and to acquire a structure that is essential for function.