  Let $X$ be a Berkovich space over a valued field. We prove that every finite
group is a Galois group over $\Ms(B)(T)$, where $\Ms(B)$ is the field of
meromorphic functions over a part $B$ of $X$ satisfying some conditions. This
gives a new geometric proof that every finite group is a Galois group over
$K(T)$, where $K$ is a complete valued field with non-trivial valuation.
  Then we switch to Berkovich spaces over ${\bf Z}$ and use a similar strategy
to give a new proof of the following theorem by D. Harbater: every finite group
is a Galois group over a field of convergent arithmetic power series. We
believe our proof to be more geometric and elementary that the original one. We
have included the necessary background on Berkovich spaces over ${\bf Z}$.
