A polynomial f(x) with rational coefficients induces
a self-map from the set of algebraic numbers to itself, and thus
determines a discrete dynamical system. Given a rational base
point b, the Galois groups of the splitting fields of the
polynomials f^n(x)-b are determined by the backward orbit
of b under repeated iteration of f in this dynamical system.
We examine the structure of these Galois groups. In particular,
we describe them using arboreal representations, i.e., their
realizations as groups of tree automorphisms. We then survey
some results and open questions in this field (pun intended).
This talk should be accessible to anyone who is familiar with
the introductory definitions and concepts of Galois Theory.