The group of $\mathbb{C}$-algebra automorphisms of $\mathbb{C}[x,y]$ is well-known, see, for example, the proof of
[Dicks](http://www.raco.cat/index.php/PublicacionsSeccioMatematiques/article/viewFile/37473/37347) or the proof of
[Mckay and Wang](http://www.sciencedirect.com/science/article/pii/0022404988901375).



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> What can be said about the group of $\mathbb{C}$-algebra automorphisms of $\mathbb{C}(x,y)$? 
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Of course, every $\mathbb{C}$-algebra automorphism of $\mathbb{C}[x,y]$ yields a $\mathbb{C}$-algebra automorphism of $\mathbb{C}(x,y)$, but there are more $\mathbb{C}$-algebra automorphisms of $\mathbb{C}(x,y)$, for example,
$x \mapsto x^{-1}, y \mapsto y$. 



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> Can one find all of them and characterize that group?
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**EDIT:** After letting me know that I am looking for the Cremona group,
I wish to quote from [wikipedia](https://en.wikipedia.org/wiki/Cremona_group):
"In two dimensions, Max Noether and Castelnuovo showed that the complex Cremona group is generated by the standard quadratic transformation, along with PGL(3, k), though there was some controversy about whether their proofs were correct, and Gizatullin (1983) gave a complete set of relations for these generators". 


What was the problem in their proof (if at all), and is there another published proof?


Thank you very much!



Finding all automorphisms of $\mathbb{C}(x,y)$
The proof of Noether that the automorphisms of $\mathbb{C}(x,y)$ are generated by $\mathrm{PGL}(3,\mathbb{C})$ and the standard quadratic transformation had troubles because of infinitely near points. The proof of Castelnuovo (1907 is completely valid and is the first one which is accurate. This is why the theorem is Called "Noether-Castelnuovo theorem". You can find a lot of proofs of this result. See for instance the book "Geometry of the Plane Cremona Maps" of Maria Alberich Carramiñana, which gives a detailed proof and explains the history of thre problem.


As YCor said, knowing the generators does not directly give a good understanding of the group. For instance, it is useless here to find conjugacy classes of elements of finite order (but the classification is known using birational geometry). There are many other results on the group that you can find in the literature.


