The organization of the genome within interphase nuclei, and how it interacts with nuclear structures is important for the regulation of nuclear functions. Many of the studies researching the importance of genome organization and nuclear structure are performed in young, proliferating, and often transformed cells. These studies do not reveal anything about the nucleus or genome in nonproliferating cells, which may be relevant for the regulation of both proliferation and replicative senescence. Here, we provide an overview of what is known about the genome and nuclear structure in senescent cells. We review the evidence that nuclear structures, such as the nuclear lamina, nucleoli, the nuclear matrix, nuclear bodies (such as promyelocytic leukemia bodies), and nuclear morphology all become altered within growth-arrested or senescent cells. Specific alterations to the genome in senescent cells, as compared to young proliferating cells, are described, including aneuploidy, chromatin modifications, chromosome positioning, relocation of heterochromatin, and changes to telomeres.