Let us say that two conjugacy classes of a group commute if they contain representatives that commute. When G is a finite group with a normal subgroup N such that G/N is cyclic, one can use this definition, together with Hall's Marriage Theorem, to describe the distribution of the conjugacy classes of G across the cosets of N. I will give an overview of this result, and then talk about some more recent work on commuting conjugacy classes in symmetric and general linear groups. This talk is on joint work with John Britnell.