  We review the notion of differential Fay identities and demonstrate, through
case studies, its new role in integrable hierarchies of the KP type. These
identities are known to be a convenient tool for deriving dispersionless Hirota
equations. We show that differential (or, in the case of the Toda hierarchy,
difference) Fay identities play a more fundamental role. Namely, they are
nothing but a generating functional expression of the full set of auxiliary
linear equations, hence substantially equivalent to the integrable hierarchies
themselves. These results are illustrated for the KP, Toda, BKP and DKP
hierarchies. As a byproduct, we point out some new features of the DKP
hierarchy and its dispersionless limit.
