The origin of eukaryotic cells is one of the most fascinating challenges in biology, and has inspired decades of controversy and debate. Recent work has led to major upheavals in our understanding of eukaryotic origins and has catalysed new debates about the roles of endosymbiosis and gene flow across the tree of life. Improved methods of phylogenetic analysis support scenarios in which the host cell for the mitochondrial endosymbiont was a member of the Archaea, and new technologies for sampling the genomes of environmental prokaryotes have allowed investigators to home in on closer relatives of founding symbiotic partners. The inference and interpretation of phylogenetic trees from genomic data remains at the centre of many of these debates, and there is increasing recognition that trees built using inadequate methods can prove misleading, whether describing the relationship of eukaryotes to other cells or the root of the universal tree. New statistical approaches show promise for addressing these questions but they come with their own computational challenges. The papers in this theme issue discuss recent progress on the origin of eukaryotic cells and genomes, highlight some of the ongoing debates, and suggest possible routes to future progress.