Automorphisms of complex affine $n$-space, old and new

The group $GA(n)$ of regular automorphisms of complexe affine $n$-space $\mathbb{A}^n$ is an important example of a so-called ind-group (or infinite dimensional algebraic group). Quite a lot is known in dimension $n=2$ due to the structure of $GA(2)$ as an amalgamated product. E.g. the unipotent elements form a closed subset, and the conjugacy classes of semisimple elements are closed. In higher dimension, most of these questions are completely open and we have only very few answers. Starting from classical results we will report on new developments and formulate a number of interesting problems.