This is a talk about work on progress on the classification of the K-theory of monoid sets, following on work of Haesemeyer--Weibel. Haesemeyer--Weibel showed that the K'-theory of monoid sets (in which they restricted attention to the "partially cancellative" monoid sets) satisfies the Fundamental Theorem of K-theory. If this restriction is omitted, it is straightforward to see that the Fundamental Theorem cannot hold. In this talk we give a description of the K-theory of finite abelian monoid sets, and discuss extensions of the Fundamental Theorem to this context. This is joint work with Mary Sarazola and Brandon Shapiro.