Research on the genomic architecture of speciation has increasingly revealed the importance of structural variants (SVs) that affect the presence, abundance, position, and/or direction of a nucleotide sequence. SVs include large chromosomal rearrangements such as fusion/fissions, inversions and translocations, as well as smaller variants such as duplications, insertions, and deletions (CNVs). Although we have ample evidence that SVs play a key role in speciation, the underlying mechanisms differ depending on the type and length of the SV, as well as the ecological, demographic and historical context. We review predictions and empirical evidence for classic processes such as underdominance due to meiotic aberrations and the coupling effect of recombination suppression before exploring how recent sequencing methodologies illuminate the prevalence and diversity of SVs. We discuss specific properties of SVs and their impact throughout the genome, highlighting that multiple processes are at play, and possibly interacting, in the relationship between SVs and speciation.