The Double Cut and Join is an operation acting locally at four chromosomal positions without regard to chromosomal context. This chapter discusses its application and the resulting menu of operations for genomes consisting of arbitrary numbers of circular chromosomes, as well as for a general mix of linear and circular chromosomes. In the general case the menu includes: inversion, translocation, transposition, formation and absorption of circular intermediates, conversion between linear and circular chromosomes, block interchange, fission, and fusion. This chapter discusses the well-known edge graph and its dual, the adjacency graph, recently introduced by Bergeron et al. Step-by-step procedures are given for constructing and manipulating these graphs. Simple algorithms are given in the adjacency graph for computing the minimal DCJ distance between two genomes and finding a minimal sorting; and use of an online tool (Mauve) to generate synteny blocks and apply DCJ is described.