This chapter develops the mathematical and algorithmic techniques utilized to address gene rearrangement problems, both in their original sense as markers of inherited traits, and in their modern interpretation as families of transcripts. This chapter focuses on the rearrangement problem between representative genomes of different species. Modern genome sequencing and annotation techniques identify the position and orientation of genes on the chromosomes. The chapter addresses the problem of enumerating the set of all optimal scenarios that transform one genome into another. The basic principle can be described in terms of balanced cycles. The chapter computes the sequences of double-cut-and-join (DCJ) operations of minimum length. One way to build balanced cycles uses the dashed edges to balance an unbalanced cycle. These dashed edges make it possible to model the fact that a DCJ operation can change the number of chromosomes, and as such, the number of telomeres in a genome, while maintaining the same gene content.