Galois groups with finite ramification over $Q$ are the "fundamental groups" of number theory. Most of what we know about them stems from their action on certain $p$-adic vector spaces. In this talk, I will describe their action on certain trees, which promises to throw a different kind of light on fundamental groups. Let $K$ be a number field and $f(x)$ in $K[x]$ be a polynomial whose critical points are preperiodic under iteration of $f$. Then every $K$-rational specialization of the tower of iterates of $f:P^1 --> P^1$ is finitely ramified. This leads to a number of open problems about the nature of the corresponding "iterated monodromy" representations of the Galois group of $K$. This is joint work with Christian Maire (Toulouse) and Wayne Aitken (CSU San Marcos).