Polynomial automorphisms, especially over finite fields

A polynomial endomorphism of $k^n$ ($k$ a field)
is $F:=(F_1,\ldots,F_n)$ where the $F_i\in k[X_1,\ldots,X_n]$.
The set of polynomial automorphisms over $k$ is a group (denoted by $\textup{GA}_n(k)$).
For $n=1$ the group consists of affine maps $x\longrightarrow ax+b$. For $n=2$ the description is a bit more complicated, but still understandable:
a certain subgroup of $\textup{GA}_n(k)$ is called the group of ``tame automorphisms'' (denoted by $\textup{TA}_n(k)$)
Now $\textup{TA}_2(k)=\textup{GA}_2(k)$, which enables the proof of many facts in dimension 2.
However, for $n\geq 3$ we're completely in the dark. The only recent result is negative: if $\textup{kar}(k)=0$, then $\textup{GA}_3(k)\not = \textup{TA}_3(k)$.
We don't even have a list of describable automorphisms which generate $\textup{GA}_n(k)$ if $n\geq 3$ !

There is some hope, though, as there are recent results on the subgroup $\textup{GA}_2(k[X_3])\subset \textup{GA}_3(k)$.

In this talk I will give a short survey on the automorphism group $\textup{GA}_n(k)$ and what we know (which is not much if $n\geq 3$).
In particular, I will discuss the case where $\textup{kar}(k)=p$, especially when $k$ is finite. Suddenly, one ends up in finite group theory,
which enables us to prove some surprising (and entertaining!) results.

I will keep things simple. If you understand ``polynomial ring'' and ``group'' then you'll understand most of what I will say.