Common bread wheat, Triticum aestivum, has one of the most complex genomes known to science, with 6 copies of each chromosome, enormous numbers of near-identical sequences scattered throughout, and an overall haploid size of more than 15 billion bases. Multiple past attempts to assemble the genome have produced assemblies that were well short of the estimated genome size. Here we report the first near-complete assembly of T. aestivum, using deep sequencing coverage from a combination of short Illumina reads and very long Pacific Biosciences reads. The final assembly contains 15 344 693 583 bases and has a weighted average (N50) contig size of 232 659 bases. This represents by far the most complete and contiguous assembly of the wheat genome to date, providing a strong foundation for future genetic studies of this important food crop. We also report how we used the recently published genome of Aegilops tauschii, the diploid ancestor of the wheat D genome, to identify 4 179 762 575 bp of T. aestivum that correspond to its D genome components.© The Author 2017. Published by Oxford University Press.