We discuss the dynamic and structural properties of polynomial semigroups, a
natural extension of iteration theory to random (walk) dynamics, where the
semigroup $G$ of complex polynomials (under the operation of composition of
functions) is such that there exists a bounded set in the plane which contains
any finite critical value of any map $g \in G$. In general, the Julia set of
such a semigroup $G$ may be disconnected, and each Fatou component of such $G$
is either simply connected or doubly connected (\cite{Su01,Su9}).