  We prove that the Euclidean ball can be realized as a Fatou component of a
holomorphic automorphism of $\mathbb{C}^m$, in particular as the escaping and
the oscillating wandering domain. Moreover, the same is true for a large class
of bounded domains, namely for all bounded regular open sets $\Omega\subset
\mathbb{C}^m$ whose closure is polynomially convex. Our result gives in
particular the first example of a bounded Fatou component with a smooth
boundary in the category of holomorphic automorphisms.
