  The objects of study in this paper are Hopf algebras $H$ which are finitely
generated algebras over an algebraically closed field and are extensions of a
commutative Hopf algebra by a finite dimensional Hopf algebra. Basic structural
and homological properties are recalled and classes of examples are listed.
Bounds are obtained on the dimensions of simple $H$-modules, and the structure
of $H$ is shown to be severely constrained when the finite dimensional
extension is semisimple and cosemisimple.
