This is a survey on the state-of-the-art of the classification of
finite-dimensional complex Hopf algebras. This general question is addressed
through the consideration of different classes of such Hopf algebras. Pointed
Hopf algebras constitute the class best understood; the classification of those
with abelian group is expected to be completed soon and there is substantial
progress in the non-abelian case.Comment: 25 pages. To be presented at the algebra session of ICM 2014.
Submitted versio