The genomes of mammalian cells exist as chromatin—a complex and dynamic structure that both serves as the background and actively participates in all fundamental nuclear processes such as transcription, replication, and DNA repair. Chromatin is characterized by multiple levels of organization: at the primary level, DNA is wrapped around a set of proteins called histones; interactions between histones promote further folding of the nucleoprotein fiber into a 30-nm structure with an unknown mechanical composition. The 30-nm fibers are then additionally folded into higher-order domains with differing structural and functional properties, which are then arranged in the nucleus in a probabilistic manner, with gene-poor regions preferring the periphery and gene-rich regions accumulating in the interior. Every level of chromatin organization has relevance for genome stability. At the 30-nm fiber level, the chromatin response to DNA damage is driven by the “access, repair, restore model,” while higher levels of organization determine the frequency and nature of chromosomal translocations. A modern view of genome stability aims to integrate the influence of fundamental cellular processes such as transcription and replication with chromatin context to give a better understanding of the processes that shaped our genomes.