Structural variation (SV) encompasses diverse types of genomic variants including deletions, duplications, inversions, transpositions, translocations, and complex rearrangements, and is now recognized to be an abundant class of genetic variation in mammals. Different individuals, or strains, of a given species can differ by thousands of variants. However, despite a large number of studies over the past decade and impressive progress on many fronts, there remain significant gaps in our knowledge, particularly in species other than human. Arguably the most relevant among these are genetically tractable models such as mouse, rat, and dog. The emergence of efficient and affordable DNA sequencing technologies presents an opportunity to make rapid progress toward understanding the nature, origin, and function of SV in these, and other, domesticated species. Here, we summarize the current state of knowledge of SV in mammals, with a focus on the similarities and differences between domesticated species and human. We then present methods to identify SV breakpoints from next-generation sequence (NGS) data by paired-end mapping, split-read mapping, and local assembly, and discuss challenges that arise when interpreting these data in lineages with complex breeding histories and incomplete reference genomes. We further describe technical modifications that allow for identification of variants involving repetitive DNA elements such as transposons and segmental duplications. Finally, we explore a few of the key biological insights that can be gained by applying NGS methods to model organisms.