The ability to save the state of a system is a fundamental aspect of many models of computation. This is also fundamental to biological systems, which use memory mechanisms for a wide variety of tasks, including producing the sustained gene expression patterns that drive cell differentiation. Prevailing strategies for engineering memory in synthetic biological systems include: 1) utilizing recombinases to invert the orientation of specific pieces of DNA (DNA rearrangement); 2) constructing positive feedback systems using transcription factors. An additional mechanism for achieving biological memory in eukaryotes is through epigenetic modifications to chromatin. We are comprehensively exploring the genetic design space for cellular memory and building synthetic memory elements in eukaryotic cells, with specific emphasis on developing chromatin-based memory elements and circuits.