Commutativity conditions for rings: 1950–2005

During the last 55 years there have been many results concerning conditions that force a ring to be commutative. These results were stimulated by Jacobsonʹs famous result and were extensively developed by Herstein. This paper will survey the area by organizing the results according to whether they come from variations on Hersteinʹs conditions, depend on general polynomial conditions, depend on the presence of a derivation, or whether a ring has special properties that make commutativity more easily accessible. Finally, the most recent conditions concern product sets and lead to results in a new area of inquiry.