Abstract: Equivalence relations play a fundamental role in the world of mathematics. One is exposed to this notion, of course unknowingly, as early as middle school when one learns about fractions. Though it is interesting and challenging, this talk, unfortunately, will not be about equivalent fractions. Instead, I will talk about measure equivalence of groups, a notion introduced by Gromov as a measurable analogue of the geometric notion of quasi-isometry of groups. I will provide a brief survey of a few results in the measure equivalence theory and its connections to the theory of von Neumann algebras. After putting on our noncommutative glasses (no need to worry if you don't own one), we will discuss the noncommutative measure equivalence (of groups), a notion introduced by Jesse Peterson, Lauren Ruth, and myself, and explore the stability of a few properties of groups under this equivalence relation.