Maintaining genome stability is essential for the accurate transmission of genetic material. Genetic instability is associated with human genome disorders and is a near-universal hallmark of cancer cells. Genetic variation is also the driving force of evolution, and a genome must therefore display adequate plasticity to evolve while remaining sufficiently stable to prevent mutations and chromosome rearrangements leading to a fitness disadvantage. A primary source of genome instability are errors that occur during chromosome replication. More specifically, obstacles to the movement of replication forks are known to underlie many of the gross chromosomal rearrangements seen both in human cells and in model organisms. Obstacles to replication fork progression destabilize the replisome (replication protein complex) and impact on the integrity of forked DNA structures. Therefore, to ensure the successful progression of a replication fork along with its associated replisome, several distinct strategies have evolved. First, there are well-orchestrated mechanisms that promote continued movement of forks through potential obstacles. Second, dedicated replisome and fork DNA stabilization pathways prevent the dysfunction of the replisome if its progress is halted. Third, should stabilisation fail, there are mechanisms to ensure damaged forks are accurately fused with a converging fork or, when necessary, re-associated with the replication proteins to continue replication. Here, we review what is known about potential barriers to replication fork progression, how these are tolerated and their impact on genome instability.