A Hopf algebra is an associative algebra, A, for which also the dual space A* carries a compatible algebra structure (plus some more axioms). We will introduce the concepts, starting from a basic algebra review. Plenty of examples will be given and axioms will be put into a categorical context. The term Hopf algebra originates form Heinz Hopf's work on the cohomology rings of Lie groups, which exhibit exactly this structure. Since then Hopf algebras have been applied in many other disciplines of mathenmatics. These include homotopy theory, representation theory, non-commutative geometry, low-dimensional topology, combinatorics/graph-theory, and even particle physics and computer science. We will discuss a few of these applications. Time permitting, we also introduce the concept of quasi-triangular Hopf algebras (aka, "Quantum Groups") - a recently discovered, very rich and useful family of Hopf algebras.

Much of the success of Hopf algebras is due to their remarkable structural rigidity. As we go along we will point out some manifestations of this, such as the Hopf/Borel Theorems, the existence of special elements (integrals, Frobenius elements, etc), as well as the Tannaka-Krein dualities.

Several examples of (co)commutative Hopf algebras will be used to explain the idea of products and coproducts as assemblers and disassemblers. We also give some popular examples of Hopf algebras that are neither commutative or cocommutative and sketch their relevance for low-dim topology.