Mammalian nuclei are equipped with a framework of intermediate filaments that function as a karyoskeleton. This nuclear scaffold, formed primarily by lamins (A-type and B-type), maintains the spatial and functional organization of the genome and of sub-nuclear compartments. Over the past decade, a body of evidence has highlighted the significance of these structural nuclear proteins in the maintenance of nuclear architecture and mechanical stability, as well as genome function and integrity. The importance of these structures is now unquestioned given the wide range of degenerative diseases that stem from LMNA gene mutations, including muscular dystrophy disorders, peripheral neuropathies, lipodystrophies, and premature aging syndromes. Here, we review our knowledge about how alterations in nuclear lamins, either by mutation or reduced expression, impact cellular mechanisms that maintain genome integrity. Despite the fact that DNA replication is the major source of DNA damage and genomic instability in dividing cells, how alterations in lamins function impact replication remains minimally explored. We summarize recent studies showing that lamins play a role in DNA replication, and that the DNA damage that accumulates upon lamins dysfunction is elicited in part by deprotection of replication forks. We also discuss the emerging model that DNA damage and replication stress are “sensed” at the cytoplasm by proteins that normally survey this space in search of foreign nucleic acids. In turn, these cytosolic sensors activate innate immune responses, which are materializing as important players in aging and cancer, as well as in the response to cancer immunotherapy.