There is a strong interplay between geometry and topology on one hand, and group theory on the other, with tools and concepts from one area being transferred and adapted to the other one. The notion of commensurability, first introduced in geometry by the Greeks, is a good example of this exchange between the two subjects.
I will discuss how the definition of commensurability has evolved to capture different features of geometry and group theory, and give several examples of groups and spaces that are commensurable in different senses. I will also discuss the relationship between commensurability and the weaker notion of quasi-isometry, again by providing a variety of examples.