  We show that the automorphism group of affine n-space $A^n$ determines $A^n$
up to isomorphism: If $X$ is a connected affine variety such that $Aut(X)$ is
isomorphic to $Aut(A^n)$ as ind-groups, then $X$ is isomorphic to $A^n$ as a
variety. We also show that every finite group and every torus appears as
$Aut(X)$ for a suitable affine variety $X$, but that $Aut(X)$ cannot be
isomorphic to a semisimple group. In fact, if $Aut(X)$ is finite dimensional
and $X$ not isomorphic to the affine line $A^1$, then the connected component
$Aut(X)^0$ is a torus.
  Concerning the structure of $Aut(A^n)$ we prove that any homomorphism
$Aut(A^n) \to G$ of ind-groups either factors through the Jacobian determinant
$jac\colon Aut(A^n) \to k^*$, or it is a closed immersion. For
$SAut(A^n):=\ker(jac)$ we show that every nontrivial homomorphism $SAut(A^n)
\to G$ is a closed immersion. Finally, we prove that every non-trivial
homomorphism $SAut(A^n) \to SAut(A^n)$ is an automorphism, and is given by
conjugation with an element from $Aut(A^n)$.
