Taking formations of groups and of inverse semigroups as the starting point, formations of orthodox semigroups are defined, as well as the wider class of i-formations (i standing for idempotent-separating).

The relation between the nature of a class of inverse semigroups $\mathcal{F}$ [of groups $\mathcal{G}$] and that of certain classes of orthodox semigroups with associated inverse semigroups in $\mathcal{F}$ [groups in $\mathcal{G}$] is discussed.

The product of formations of orthodox semigroups, in particular of $R$-unipotent semigroups, is considered, and a product like the Gasch{\"u}tz´s product known for groups is presented for i-formations.

The paper concludes with a list of questions.

CEMAT - Center for Computational and Stochastic Mathematics