Rearrangements of the genome can be detected by microarray methods and massively parallel sequencing, which identify copy-number alterations and breakpoint junctions, but these techniques are poorly suited to reconstructing the long-range organization of rearranged chromosomes, for example, to distinguish between translocations and insertions. The single-DNA-molecule technique HAPPY mapping is a method for mapping normal genomes that should be able to analyse genome rearrangements, i.e. deviations from a known genome map, to assemble rearrangements into a long-range map. We applied HAPPY mapping to cancer cell lines to show that it could identify rearrangement of genomic segments, even in the presence of normal copies of the genome. We could distinguish a simple interstitial deletion from a copy-number loss at an inversion junction, and detect a known translocation. We could determine whether junctions detected by sequencing were on the same chromosome, by measuring their linkage to each other, and hence map the rearrangement. Finally, we mapped an uncharacterized reciprocal translocation in the T-47D breast cancer cell line to about 2 kb and hence cloned the translocation junctions. We conclude that HAPPY mapping is a versatile tool for determining the structure of rearrangements in the human genome.