Recent work has emphasised that the human genome is not simple and static, but complex and dynamic. This review focuses on the regions that are particularly hard to dissect and analyse, yet hold clues to how the genome changes during evolution and disease. I begin by summarising recent key advances in the understanding of the variable structure of our genome, and then I discuss a medley of methods that may allow us to analyse this structure in fine detail. In the final part, I describe potential future developments in this field, and make an argument that, just as we routinely genotype single-nucleotide polymorphisms now and will routinely re-sequence genomes in the near future, we should be aiming to physically re-map the individual human genome for each individual we study.