Integrals for Hopf algebras and their relation to the antipode

We will give a brief introduction to integrals on Hopf algebras, review some results about their connection with the antipode, and illustrate how the mere existence of integrals brings some sort of finiteness properties to the Hopf algebra. The talk will be accessible to a wide audience, there will be no prerequisites, and all notions involved will be explained and illustrated by lots of examples. Towards the end of the talk we will mention recent results obtained jointly with Margaret Beattie and Miodrag Iovanov.