  We develop a $p$-adic version of the so-called Grothendieck-Teichm\"uller
theory (which studies $Gal(\bar{\bf Q}/{\bf Q})$ by means of its action on
profinite braid groups or mapping class groups). For every place $v$ of
$\bar{\bf Q}$, we give some geometrico-combinatorial descriptions of the local
Galois group $Gal(\bar{\bf Q}_v/{\bf Q}_v)$ inside $Gal(\bar{\bf Q}/{\bf Q})$.
We also show that $Gal(\bar{\bf Q}_p/{\bf Q}_p)$ is the automorphism group of
an appropriate $\pi_1$-functor in $p$-adic geometry.
