Hypotheses about the origin of eukaryotic cells are classically framed within the context of a universal 'tree of life' based on conserved core genes. Vigorous ongoing debate about eukaryote origins is based on assertions that the topology of the tree of life depends on the taxa included and the choice and quality of genomic data analysed. Here we have reanalysed the evidence underpinning those claims and apply more data to the question by using supertree and coalescent methods to interrogate >3,000 gene families in archaea and eukaryotes. We find that eukaryotes consistently originate from within the archaea in a two-domains tree when due consideration is given to the fit between model and data. Our analyses support a close relationship between eukaryotes and Asgard archaea and identify the Heimdallarchaeota as the current best candidate for the closest archaeal relatives of the eukaryotic nuclear lineage.