Reference about $k$-algebra isomorphism of $k[x_1,\ldots,x_n]$ $k$ is a field, we can assume $k$ is algebraic closed if we need.
I donot know if the construction of $\mathrm{Aut}_{k-alg}k[X_1,\ldots,X_n]$ is known.
I donot know any references about this group $\mathrm{Aut}_{k-alg}k[X_1,\ldots,X_n]$.

What do we know about the group so far?
  Could anyone give some references about this?

Thanks.
 A: The group is not known, except when $n=1$ and when $n=2$. The first case is very easy to handle. The second one is already quite non-trivial, but one can see that the group is the amalgamated product of the linear group and the group of triangular automorphisms (those of the form $x\mapsto x$, $y\mapsto y+f(x)$, for some $f\in k[x]$).
There is a great book by Arno van den Essen which deals with this problem and related ones, like the Jacobian conjecture.
Very recently, though, Shestakov and Umirbaev were able to show that the group of automorphisms of $k[x,y,z]$ is not generated by the linear automorphisms and the triangular automorphisms. They in fact proved that an explicit example of an automorphism, proposed by Nagata years ago, is wild. 
van den Essen's book has an up-to-date bibliography.
 
There are very closedly related problems that are also of great interest. For example, the group of automorphisms of Weyl algebras $A_n$ are also not known except when $n=1$, where Alev and Chamarie obtained a description similar to that of $Aut(k[x,y])$. There is an amazing conjecture by Kontsevich about the form of the group in this case---google should find it in no time. 
Similarly, people have looked at $Aut$ of enveloping algebras of Lie algebras, of primitive quotients of enveloping algebras of (mostly semisimple) Lie algebras, which generalize Weyl algebras, of the quantum variants of these, and so on. What everyone wants is polynomial rings, though :)
