The recent coronavirus disease 2019 (COVID-19) pandemic was caused by severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2). COVID-19 is characterized by respiratory distress, multiorgan dysfunction and, in some cases, death. The virus is also responsible for post-COVID-19 condition (commonly referred to as ‘long COVID’). SARS-CoV-2 is a single-stranded, positive-sense RNA virus with a genome of approximately 30 kb, which encodes 26 proteins. It has been reported to affect multiple pathways in infected cells, resulting, in many cases, in the induction of a ‘cytokine storm’ and cellular senescence. Perhaps because it is an RNA virus, replicating largely in the cytoplasm, the effect of SARS-Cov-2 on genome stability and DNA damage responses (DDRs) has received relatively little attention. However, it is now becoming clear that the virus causes damage to cellular DNA, as shown by the presence of micronuclei, DNA repair foci and increased comet tails in infected cells. This review considers recent evidence indicating how SARS-CoV-2 causes genome instability, deregulates the cell cycle and targets specific components of DDR pathways. The significance of the virus’s ability to cause cellular senescence is also considered, as are the implications of genome instability for patients suffering from long COVID.