The genomes of closely related individuals or species differ from one another by several different classes of mutations. The most commonly studied class of mutations are single nucleotide substitutions. However, it is also interesting to study larger mutations such as duplications or inversions of large stretches of DNA because such mutations are rarer and therefore allow us to study more distant evolutionary relationships. Since they are larger they may also be more likely to affect phenotypic traits. In this talk, I will present theoretical results for combinatorial problems that arise from the study of genome rearrangements. I will also show recent experimental results on real data from the fruit fly, Drosophila melanogaster.