There is an increasing understanding that variation in gene presence-absence plays an important role in the heritability of agronomic traits; however, there have been relatively few studies on variation in gene presence-absence in crop species. Hexaploid wheat is one of the most important food crops in the world and intensive breeding has reduced the genetic diversity of elite cultivars. Major efforts have produced draft genome assemblies for the cultivar Chinese Spring, but it is unknown how well this represents the genome diversity found in current modern elite cultivars. In this study we build an improved reference for Chinese Spring and explore gene diversity across 18 wheat cultivars. We predict a pangenome size of 140 500 ± 102 genes, a core genome of 81 070 ± 1631 genes and an average of 128 656 genes in each cultivar. Functional annotation of the variable gene set suggests that it is enriched for genes that may be associated with important agronomic traits. In addition to variation in gene presence, more than 36 million intervarietal single nucleotide polymorphisms were identified across the pangenome. This study of the wheat pangenome provides insight into genome diversity in elite wheat as a basis for genomics-based improvement of this important crop. A wheat pangenome, GBrowse, is available at http://appliedbioinformatics.com.au/cgi-bin/gb2/gbrowse/WheatPan/, and data are available to download from http://wheatgenome.info/wheat_genome_databases.php.