F. J. Murray and J. von Neumann introduced the family of unbounded, closed, densely-defined operators that are closely associated (“affiliated” as they termed them) with a finite von Neumann algebra. They proved that they have remarkable common, dense domain properties and described surprising addition and multiplication operations for them. Zhe Liu and the speaker have defined algebras based on these operations, Murray-von Neumann algebras, and studied their basic structure. This will be discussed during the lecture with special emphasis on the nature of the derivations of these algebras.