One of the miracles of the theory of connected reductive groups over fields is that every such group has a unique "split" form for the étale topology, and these split forms are classified by combinatorial data independent of the ground field. Informally, connected reductive groups over fields "have no moduli" for the étale topology.
The generic fiber of a family of more general smooth connected affine groups over finite or algebraically closed ground fields of positive characteristic generally cannot be related to a reductive group because its geometric unipotent radical can fail to be defined over the (imperfect) function field of the parameter space. The theory of pseudo-reductive groups was developed several years ago (with O. Gabber and G. Prasad) in order to overcome this problem, and such groups turned out to "vary with moduli" in a mild way that could be completely understood except when [k:k2] > 2 with char(k)=2 (a degree restriction that is harmless for many arithmetic purposes but is ultimately not satisfactory).
If [k:k2] > 2 with char(k)=2 then there is an explosion in the possibilities for pseudo-reductive groups: they can "vary with moduli" in ways that are far richer than in any other case. In this talk, after some motivation for and overview of the general concepts involved, I will show lots of examples of such groups, each related to a special phenomenon in the combinatorial theory of root systems. In a sense I will make precise, the list of examples is exhaustive. This is joint work with G. Prasad.