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. a 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 examples and 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.