Let $A$ be a finite dimensional algebra over a field $K$. $A$ is called periodic if $A$ as an $A$-bimodule is a periodic module, that is $\Omega^n(A) \cong A$ for some $n \geq 1$. Being periodic implies that the algebra is selfinjective.
Question: Is a classification of periodic Hopf algebras known?
For group algebras the answer is very nice: When $KG$ is periodic for all algebraically closed fields $K$ if and only if $G$ has the property that every abelian subgroup is cyclic.
Other examples of periodic Hopf algebras are Taft algebras.
I do not know any other non-trvial examples (trivial examples are semisimple Hopf algebras over perfect fields).