Many of our major crop species are polyploids, containing more than one genome or set of chromosomes. Polyploid crops present unique challenges, including difficulties in genome assembly, in discriminating between multiple gene and sequence copies, and in genetic mapping, hindering use of genomic data for genetics and breeding. Polyploid genomes may also be more prone to containing structural variation, such as loss of gene copies or sequences (presence–absence variation) and the presence of genes or sequences in multiple copies (copy-number variation). Although the two main types of genomic structural variation commonly identified are presence–absence variation and copy-number variation, we propose that homeologous exchanges constitute a third major form of genomic structural variation in polyploids. Homeologous exchanges involve the replacement of one genomic segment by a similar copy from another genome or ancestrally duplicated region, and are known to be extremely common in polyploids. Detecting all kinds of genomic structural variation is challenging, but recent advances such as optical mapping and long-read sequencing offer potential strategies to help identify structural variants even in complex polyploid genomes. All three major types of genomic structural variation (presence–absence, copy-number, and homeologous exchange) are now known to influence phenotypes in crop plants, with examples of flowering time, frost tolerance, and adaptive and agronomic traits. In this review, we summarize the challenges of genome analysis in polyploid crops, describe the various types of genomic structural variation and the genomics technologies and data that can be used to detect them, and collate information produced to date related to the impact of genomic structural variation on crop phenotypes. We highlight the importance of genomic structural variation for the future genetic improvement of polyploid crops.