Abstract: An integral on a Hopf algebra H is a linear functional ∫ on H with an invariance property analogous to that of the Haar measure on a locally compact group. Not every Hopf algebra admits a non-zero (left) integral; those that do are called co-Frobenius. Two main early examples are: nite dimensional Hopf algebras and cosemisimple Hopf algebras. There are several characterizations of co-Frobenius Hopf algebras in cohomological terms, e.g. the existence of projectives in the category of comodules. In this talk I will report some results on co-Frobenius Hopf algebras (jointly with Cuadra and Etingof):

    * A structure theorem in terms of the so-called Hopf coradical.

    * The coradical filtration of a co-Frobenius Hopf algebra is finite, proving a conjecture by Dascalescu and myself.

* The construction of a new family of co-Frobenius Hopf algebras, a sort of blowing-up of a dual of a lifting of a quantum line; these examples provide a negative answer to a question by Dascalescu and myself.