In these lectures I will explain the basic intuitive idea behind noncommutative geometry, in the sense of A. Connes, as a duality between geometric and algebraic objects. Many notions and invariants of geometric and topological nature have their counterparts in noncommutative geometry. In particular cyclic homology is the noncommutative analogue of de Rham cohomology of smooth manifolds. It pairs with both topological and algebraic K-theory of noncommutative algebras. I will explain how a special class of noncommutative and non-cocommutative Hopf algebras naturally appeared in noncommutative geometry in the work of Connes and Moscovici on local index formula for foliated manifolds. The last parts of these lectures will be devoted to current work on the cyclic cohomology of Hopf algebras. These lectures are meant to be a gentel introduction to the ideas involved and should be accessible to graduate students and postdoctoral fellows and those who are interested. I will make sure to provide the necessary background material during the lectures.