Comparisons of human genomes show that more base pairs are altered as a result of structural variation — including copy number variation — than as a result of point mutations. Here we review advances and challenges in the discovery and genotyping of structural variation. The recent application of massively parallel sequencing methods has complemented microarray-based methods and has led to an exponential increase in the discovery of smaller structural-variation events. Some global discovery biases remain, but the integration of experimental and computational approaches is proving fruitful for accurate characterization of the copy, content and structure of variable regions. We argue that the long-term goal should be routine, cost-effective and high quality de novo assembly of human genomes to comprehensively assess all classes of structural variation.