Most cancers have lost a critical DNA damage response (DDR) pathway during tumor evolution. These alterations provide a useful explanation for the initial sensitivity of tumors to DNA-targeting chemotherapy. A striking example is dysfunctional homology-directed repair (HDR), e.g., due to inactivating mutations in BRCA1 and BRCA2 genes. Extensive efforts are being made to develop novel targeted therapies exploiting such an HDR defect. Inhibitors of poly(ADP-ribose) polymerase (PARP) are an instructive example of this approach. Despite the success of PARP inhibitors, the presence of primary or acquired therapy resistance remains a major challenge in clinical oncology. To move the field of precision medicine forward, we need to understand the precise mechanisms causing therapy resistance. Using preclinical models, various mechanisms underlying chemotherapy resistance have been identified. Restoration of HDR seems to be a prevalent mechanism but this does not explain resistance in all cases. Interestingly, some factors involved in DNA damage response (DDR) have independent functions in replication fork (RF) biology and their loss causes RF instability and therapy sensitivity. However, in BRCA-deficient tumors, loss of these factors leads to restored stability of RFs and acquired drug resistance. In this review we discuss the recent advances in the field of RF biology and its potential implications for chemotherapy response in DDR-defective cancers. Additionally, we review the role of DNA damage tolerance (DDT) pathways in maintenance of genome integrity and their alterations in cancer. Furthermore, we refer to novel tools that, combined with a better understanding of drug resistance mechanisms, may constitute a great advance in personalized diagnosis and therapeutic strategies for patients with HDR-deficient tumors.