Structural variants (SVs) account for the majority of base pair differences both within and between primate species. However, our understanding of inter- and intra-species SV has been historically hampered by the quality of draft primate genomes and the absence of genome resources for key taxa. Recently, advances in long-read sequencing and genome assembly have begun to radically reshape our understanding of SVs. Two landmark achievements include the publication of a human telomere-to-telomere (T2T) genome as well as the development of the first human pangenome reference. In this review, we first look back to the major works laying the foundation for these projects. We then examine the ways in which T2T genome assemblies and pangenomes are transforming our understanding of and approach to primate SV. Finally, we discuss what the future of primate SV research may look like in the era of T2T genomes and pangenomics.