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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.