Stone-Wales (SW) rotations represent class of topological defects and topological modifications—the topological defects may actually detect in the honeycomb structure of the graphene layers and, from a formal point of view, a large class of topological modifications applicable to three-connected networks such as sp2-carbon nanoclusters, fullerenes, carbon nanotubes, graphene, and others. The “original” SW rearrangement swaps the position of two pentagons and two hexagons on the fullerene surface. The elegance of the SW transformations represented in this chapter somehow reflects the inherent and profound beauty of the topological methods applied to chemistry. SW rotations are effective graphical tools for generating valid isomers of carbon nanostructures. Their relevance is far to be fully understood, being the topic still largely unexplored. In other scientific sectors, polygonal tessellations of 3-connected systems are quite “popular” in Nature, where biological tissues especially do provide multiple examples of tessellations of cells whose topology is modifiable by SW rearrangements.