Sustaining cell identity and function across cell division is germane to human development, healthspan, and cancer avoidance. This relies significantly on propagation of chromatin organization between cell generations, as chromatin presents a barrier to cell fate and cell state conversions. Inheritance of chromatin states across the many cell divisions required for development and tissue homeostasis represents a major challenge, especially because chromatin is disrupted to allow passage of the DNA replication fork to synthesize the two daughter strands. This process also leads to a twofold dilution of epigenetic information in histones, which needs to be accurately restored for faithful propagation of chromatin states across cell divisions. Recent research has identified distinct multilayered mechanisms acting to propagate epigenetic information to daughter strands. Here, we summarize key principles of how epigenetic information in parental histones is transferred across DNA replication and how new histones robustly acquire the same information postreplication, representing a core component of epigenetic cell memory.