Complete and accurate DNA replication requires the progression of replication forks through DNA damage, actively transcribed regions, structured DNA and compact chromatin. Recent studies have revealed a remarkable plasticity of the replication process in dealing with these obstacles, which includes modulation of replication origin firing, of the architecture of replication forks, and of the functional organization of the replication machinery in response to replication stress. However, these specialized mechanisms also expose cells to potentially dangerous transactions while replicating DNA. In this Review, we discuss how replication forks are actively stalled, remodelled, processed, protected and restarted in response to specific types of stress. We also discuss adaptations of the replication machinery and the role of chromatin modifications during these transactions. Finally, we discuss interesting recent data on the relevance of replication fork plasticity to human health, covering its role in tumorigenesis, its crosstalk with innate immunity responses and its potential as an effective cancer therapy target.