Genome structural variation (SV) is a major source of genetic diversity in mammals and a hallmark of cancer. Although SV is typically defined by its canonical forms (duplication, deletion, insertion, inversion and translocation), recent breakpoint mapping studies have revealed a surprising number of 'complex' variants that evade simple classification. Complex SVs are defined by clustered breakpoints that arose through a single mutation but cannot be explained by one simple end-joining or recombination event. Some complex variants exhibit profoundly complicated rearrangements between distinct loci from multiple chromosomes, whereas others involve more subtle alterations at a single locus. These diverse and unpredictable features present a challenge for SV mapping experiments. Here, we review current knowledge of complex SV in mammals, and outline techniques for identifying and characterizing complex variants using next-generation DNA sequencing.