Cellular senescence is a state of permanent replicative arrest that allows cells to stay viable and metabolically active but resistant to apoptotic and mitogenic stimuli. Specific, validated markers can identify senescent cells, including senescence-associated β galactosidase activity, chromatin alterations, cell morphology changes, activated p16- and p53-dependent signaling and permanent cell cycle arrest. Senescence is a natural consequence of DNA replication-associated telomere erosion, but can also be induced prematurely by telomere-independent events such as failure to repair DNA double strand breaks. Here, we review the molecular pathways of senescence onset, focussing on the changes in chromatin organization that are associated with cellular senescence, particularly senescence-associated heterochromatin foci formation. We also discuss the altered dynamics of the DNA double strand break response within the context of aging cells. Appreciating how, mechanistically, cellular senescence is induced, and how changes to chromatin organization and DNA repair contributes to this, is fundamental to our understanding of the normal and premature human aging processes associated with loss of organ and tissue function in humans.