Genome evolution on the basis of gene order has been modeled by a variety of rearrangement operations such as inversions, translocations, transpositions, block interchanges, fusions and fissions. The double cut and join (DCJ) operation, introduced by Yancopoulos et al. (2005), allows to model all these classical operations while allowing for very simple algorithms, compared to earlier ones for more restricted models. In this talk, we present several of our algorithmic results related to genome rearrangement based on the DCJ model and also address some interesting mathematical modeling questions.