Commutative-by-finite Hopf algebras

Roughly speaking, a commutative-by-finite Hopf algebra is a Hopf
algebra which is an extension of a commutative Hopf algebra by a
finite dimensional Hopf algebra.
There are many big and significant classes of such algebras
(beyond of course the commutative ones and the finite dimensional ones!).
I'll make the definition precise, discuss examples
and review results, some old and some new.
No previous knowledge of Hopf algebras is necessary.