Classification of finite group schemes over a field What is known about the classification of finite group schemes over a field? By a finite group scheme I mean $Spec A$ where $A$ is a finite-dimensional algebra over a field.
Is there a full classification in case of algebraically closed field? 
 A: This problem is wide open.  Just how wide will depend on your definition of "classification".  For example, over algebraically closed fields of characteristic zero, this is equivalent to the problem of classifying finite groups up to isomorphism.  We normally say that the problem of classifying finite groups up to isomorphism is a wide open problem that will never be solved, but you might be satisfied to stop once the equivalence is given.
Finite group schemes satisfy the same Jordan-Hölder property as finite groups, so you can reduce the problem to the 2-step Hölder program that we see in the finite group world:


*

*classifying finite simple group schemes

*the extension problem: If you have $G_1$ arbitrary and $G_3$ simple, classify the  groups $G_2$ with $G_1 \triangleleft G_2$ and $G_2/G_1 \cong G_3$.


The first part is done in a weak sense, by a combination of Oort-Tate for order $p$ and the fact that non-abelian simple group schemes are étale.  Rather, it is definitely done over algebraically closed fields, where it is basically the classification of finite simple groups, and then one has to classify homomorphisms from the absolute Galois group of $k$ to the automorphism group of a finite simple group (which will never be done for general $k$).
The extension problem is even more hopeless than in the finite group world.  In characteristic $p$, you have to deal with the substantial additional complication of infinitesimal structure, and even in the étale case, the descent data seem to mix in some complicated way.
