  This is primarily a survey of the way in which Hopf cyclic cohomology has
emerged and evolved, in close relationship with the application of the
noncommutative local index formula to transverse index theory on foliations.
Being Diff-invariant, the geometric framework that allowed us to treat the
`space of leaves' of a general foliation provides a `background independent'
set-up for geometry that could be of relevance to the handling of the the
background independence problem in quantum gravity. With this potential
association in mind, we have added some new material, which complements the
original paper and is also meant to facilitate its understanding. Section 2
gives a detailed description of the Hopf algebra that controls the `affine'
transverse geometry of codimension $n$ foliations, and Section 5 treats the
relative version of Hopf cyclic cohomology in full generality, including the
case of Hopf pairs with noncompact isotropy.
