In evolutionary biology, a genome can be represented by a permutation, with the entries of the permutation playing the role of genes. The evolutionary distance between two species can be defined as the minimum number of transformations of a certain kind that can turn the genome of one species into the genome of another.

It is easy to reduce this distance-measuring problem into a sorting problem. That is, given a permutation, let us turn it into the identity permutation using some allowed steps. This raises a family of questions that are interesting, difficult, and lead to very surprising discoveries. They would be interesting even without the biological motivation as they touch on remote-looking areas of combinatorics, group theory, and algebra.

In this talk, we will survey results and intriguing open problems that are typical in this field. Only an undergraduate knowledge of combinatorics will be assumed.