Cellular developments are programmed through various cell intrinsic and extrinsic cell signals. Understanding how cell fates are determined by combinations of cell signaling events is one of the central questions in biology with far-reaching consequences for human health and regenerative medicine. While we have technologies to profile and quantify diverse cell states, we currently lack a method to track each cell through its developmental path, observing how a sequence of cell signaling events results in the final cell state. To address these limitations, we strive to develop genomic recording technology, a third technological paradigm that converts past cellular events into changes in genomic sequences that can be later used to reconstruct the history of each cell. We first introduce a synthetic circuit architecture called ENGRAM, which converts transcriptional activation signals into precision genome editing events with the possibility of recording tens to hundreds of transcriptional activation signals within a single experiment. Next, we introduce a genomic recording platform called DNA Typewriter, which allows the reconstruction of temporal orders in genomic recordings. Finally, we apply the DNA Typewriter method to record clonal lineage relationships among cells within a developmental organoid called gastruloid, revealing how the past state of mouse embryonic stem cells may affect the fate decision of each organoid during induction.