In addition to point mutations, larger-scale structural changes (including rearrangements, duplications, insertions, and deletions) are also prevalent between different mammalian genomes. Capturing these large-scale changes is critical to unraveling the history of mammalian evolution in order to better understand the human genome. It also has profound biomedical significance, because many human diseases are associated with structural genomic aberrations. The increasing number of mammalian genomes being sequenced as well as the recent advancement in DNA sequencing technologies are allowing us to identify these structural genomic changes with vastly greater accuracy. However, there are a considerable number of computational challenges related to these problems. In this article, we introduce the ancestral genome reconstruction problem, which enables us to explain the large-scale genomic changes between species in an evolutionary context. The application of these methods to within-species structural variation and disease genome analysis is also discussed. The target audience of this article is advanced undergraduate students in biology.