Jointly with Rafael Andrist we have recently shown that an affine variety $X$ is determined, up to base field automorphisms, by the abstract semigroup of endomorphisms, provided $X$ contains a copy of the affine line. An more interesting question is how much information about $X$ can be retrieved from the group $Aut(X)$ of automorphisms of $X$. This group has the structure of an ind-group, i.e. an infinite dimensional algebraic group, a concept introduced by Shafarevich in 1966. It was recently studied by several authors, in particular in the case of affine $n$-space $A^n$. However, not much is known about this group in general, but there are a number of very interesting conjectures. In connection with the question above, we can prove the following. Theorem. If $X$ is a connected affine variety such that $Aut(X)$ is isomorphic to $Aut(A^n)$ as an ind-group, then $X$ is isomorphic to $A^n$ as a variety. We will explain these concepts and results, and describe some recent development.