The subject of this talk will be Hopf algebras and their dual theory. We will mostly focus on a particular class of Hopf algebras: noetherian Hopf algebras that are finitely-generated modules over some commutative normal Hopf subalgebra. Some properties and examples of these Hopf algebras will be mentioned. Furthermore, we will see some results on the dual of this class of Hopf algebras, some of its properties, decompositions e maybe some interesting Hopf subalgebras.