I assume that the classification of (certain families of) Hopf algebras is still an open problem, am I right?
My question is the following: What is the current state of the art? What is known about the classification of certain families of Hopf algebras? (To be more precise, say for example that I am interested in finite-dimensional or combinatorial Hopf algebras.)
The general problem about the classification of finite-dimensional Hopf algebras (over $\mathbb{C}$) is widely open. I mention some general results.
Here
Zhu, Yongchang. Hopf algebras of prime dimension. Internat. Math. Res. Notices 1994, no. 1, 53--59. MR1255253 (94j:16072), link
it is proved that a Hopf algebra of prime dimension is isomorphic to a group algebra. Here
Ng, Siu-Hung. Non-semisimple Hopf algebras of dimension $p^2$. J. Algebra 255 (2002), no. 1, 182--197. MR1935042 (2003h:16067), link
it is proved that the only Hopf algebras of dimension $p^2$ are the group algebras and the Taft algebras. Hopf algebras of dimension $2p^2$ were also classified by Hilgemann and Ng, see the following paper:
Hilgemann, Michael; Ng, Siu-Hung. Hopf algebras of dimension $2p^2$. J. Lond. Math. Soc. (2) 80 (2009), no. 2, 295--310. MR2545253 (2010h:16080), link
Of course, these are not the only general results. For a good accound related to the classification of Hopf algebras of a given dimension you may want to check the following papers:
Beattie, Margaret. A survey of Hopf algebras of low dimension. Acta Appl. Math. 108 (2009), no. 1, 19--31. MR2540955 (2010i:16054), link
M. Beattie and G. A. García. Classifying Hopf algebras of a given dimension. Preprint: arXiv:1206.6529
It is important to mention that it is very interesting to study the classification of certain families of finite-dimensional Hopf algebras. If this is your interest, maybe google can help.
For the "state-of-the-art" on combinatorial Hopf algebras, I would look at:
and in particular the 800 page book which he and Swapneel Mahajan have written entitled "Monoidal functors, species and Hopf algebras". I don't actually work on these myself though, so perhaps others will have further more precise suggestions.
For finite dimensional pointed Hopf algebras you can find a very accurate description of the state of the art in
Andruskiewitsch, Nicolás(RA-CRDBM); Schneider, Hans-Jürgen(D-MNCH-MI) On the classification of finite-dimensional pointed Hopf algebras. (English summary) Ann. of Math. (2) 171 (2010), no. 1, 375–417.
(for Hopf algebras with commutative group of group-likes) and
Andruskiewitsch, N.(RA-CRDBM-CI); Fantino, F.(RA-CRDBM-CI); Graña, M.(RA-UBAS); Vendramin, L. Finite-dimensional pointed Hopf algebras with alternating groups are trivial. (English summary) Ann. Mat. Pura Appl. (4) 190 (2011), no. 2, 225–245.
(for pointed Hopf algebras with non-commutative group of group likes).
A short summary of the situation is that for the commutative case the classification is done for every group with order with prime divisors greater than seven, for the non commutative case the situation is still wide open, there are a few example on $\mathbb{S}_{n}$ with n smaller than 6, an a lot of negative results for almost all simple groups, a part from a list of cases still open. In addition to that the classification is complete for some small degrees and for special cases of the factorization of the dimension.
