The automorphism group of $k[x_1,x_2]$ (the polynomial ring in two variables over a field $k$) is pretty well understood classically, but very little is known about the automorphism group of $k[x_1,\ldots,x_n]$ for $n\geq 3$: see "The Tame and the Wild Automorphisms of Polynomial Rings in Three Variables," by Shestakov--Umirbaev (https://doi.org/10.1090/S0894-0347-03-00440-5). See also the Wikipedia page https://en.wikipedia.org/wiki/Cremona_group for the related problem on the ring of rational functions.

The automorphism group of $k[x_1,x_2]$ (the polynomial ring in two variables over a field $k$) is pretty well understood classically, but very little is known about the automorphism group of $k[x_1,\dotsc,x_n]$ for $n\geq 3$: see "The Tame and the Wild Automorphisms of Polynomial Rings in Three Variables," by Shestakov–Umirbaev. See also the Wikipedia page for the related problem on the ring of rational functions.